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This answer is not really different from David's or Marc's; I just want to boil things down.

In brief:

If you want "weak equivalence of simplicial sheaves" to mean the same thing as "induce weak equivalence on stalks", you must localize simplicial presheaves with respect to hypercovers.

You could only localize with respect to covers, but then it may be possible for there to be an object whose stalks are all weakly contractible, but which is not itself weakly equivalent to the terminal object.

Note: a "weak equivalence on stalks" amounts to the same thing as "isomorphism of homotopy-group sheaves", so I could have instead said: if you only localize with respect to covers, then it may be possible for there to be an object with trivial homotopy-group sheaves, but which is itself not weakly equivalent to the terminal object.

I highly recommend the paper of Dugger, Hollander, and Isaksen, Hypercovers and simplicial presheaves, which explains this, and gives an example exhibiting the phenomenon I just mentioned.

Added.

For "localize with respect to covers", I mean: from an open cover of an open set $X$ build a simplicial object $C_\bullet$ of presheaves, with $C_0=\coprod U_i$, $C_1=\coprod U_i\cap U_j$, etc. Then a simplicial presheaf $F$ has descent for covers if $$F(X) \to \mathrm{holim} C_\bullet$$ is a weak equivalence of simplicial sets (for all open covers $\{U_i\}$ of open sets $X$). Note that I'm taking a homotopy limit of a cosimplicial diagram here.

If you just use the "first two steps of the diagram", you would be saying that the map $F(X)$ to the "homotopy equalizer" of $F(C_0) \rightrightarrows F(C_1)$ is a weak equivalence. This is a different condition, and is the wrong way.

Here is one way to see why it is wrong (I'll be a bit imprecise here): start with the constant presheaf whose value is an Eilenberg-MacLane space $K(G,n)$, and "localize" it to get a simplicial presheaf $F$ which has descent for covers. I want to be able to say that $F(X)$ has something to do with the cohomology of $X$. With the correct definition (full simplicial diagram), you can see that you are seeing something that looks similar to the classical definition of Cech cohomology. In fact, if $\{U_i\}$ is a "good cover", so that all finite intersections are either empty or contractible, this is what you get. If you just use the "two-step" definition, it's not clear that $F(X)$ will know anything about 3-fold intersections of open sets in the cover, so presumably can't compute Cech cohomology.

This answer is not really different from David's or Marc's; I just want to boil things down.

In brief:

If you want "weak equivalence of simplicial sheaves" to mean the same thing as "induce weak equivalence on stalks", you must localize simplicial presheaves with respect to hypercovers.

You could only localize with respect to covers, but then it may be possible for there to be an object whose stalks are all weakly contractible, but which is not itself weakly equivalent to the terminal object.

Note: a "weak equivalence on stalks" amounts to the same thing as "isomorphism of homotopy-group sheaves", so I could have instead said: if you only localize with respect to covers, then it may be possible for there to be an object with trivial homotopy-group sheaves, but which is itself not weakly equivalent to the terminal object.

I highly recommend the paper of Dugger, Hollander, and Isaksen, Hypercovers and simplicial presheaves, which explains this, and gives an example exhibiting the phenomenon I just mentioned.

Added.

For "localize with respect to covers", I mean: from an open cover of an open set $X$ build a simplicial object $C_\bullet$ of presheaves, with $C_0=\coprod U_i$, $C_1=\coprod U_i\cap U_j$, etc. Then a simplicial presheaf $F$ has descent for covers if $$F(X) \to \mathrm{holim} F(C_\bullet)$$ is a weak equivalence of simplicial sets (for all open covers $\{U_i\}$ of open sets $X$). Note that I'm taking a homotopy limit of a cosimplicial diagram here.

If you just use the "first two steps of the diagram", you would be saying that the map $F(X)$ to the "homotopy equalizer" of $F(C_0) \rightrightarrows F(C_1)$ is a weak equivalence. This is a different condition, and is the wrong way.

Here is one way to see why it is wrong (I'll be a bit imprecise here): start with the constant presheaf whose value is an Eilenberg-MacLane space $K(G,n)$, and "localize" it to get a simplicial presheaf $F$ which has descent for covers. I want to be able to say that $F(X)$ has something to do with the cohomology of $X$. With the correct definition (full simplicial diagram), you can see that you are seeing something that looks similar to the classical definition of Cech cohomology. In fact, if $\{U_i\}$ is a "good cover", so that all finite intersections are either empty or contractible, this is what you get. If you just use the "two-step" definition, it's not clear that $F(X)$ will know anything about 3-fold intersections of open sets in the cover, so presumably can't compute Cech cohomology.

This answer is not really different from David's or Marc's; I just want to boil things down.

In brief:

If you want "weak equivalence of simplicial sheaves" to mean the same thing as "induce weak equivalence on stalks", you must localize simplicial presheaves with respect to hypercovers.

You could only localize with respect to covers, but then it may be possible for there to be an object whose stalks are all weakly contractible, but which is not itself weakly equivalent to the terminal object.

Note: a "weak equivalence on stalks" amounts to the same thing as "isomorphism of homotopy-group sheaves", so I could have instead said: if you only localize with respect to covers, then it may be possible for there to be an object with trivial homotopy-group sheaves, but which is itself not weakly equivalent to the terminal object.

I highly recommend the paper of Dugger, Hollander, and Isaksen, Hypercovers and simplicial presheaves, which explains this, and gives an example exhibiting the phenomenon I just mentioned.

Added.

For "localize with respect to covers", I mean: from an open cover of an open set $X$ build a simplicial object $C_\bullet$ of presheaves, with $C_0=\coprod U_i$, $C_1=\coprod U_i\cap U_j$, etc. Then a simplicial presheaf $F$ has descent for covers if $$F(X) \to \mathrm{holim} C_\bullet$$ is a weak equivalence of simplicial sets (for all open covers $\{U_i\}$ of open sets $X$). Note that I'm taking a homotopy limit of a cosimplicial diagram here.

This answer is not really different from David's or Marc's; I just want to boil things down.

In brief:

If you want "weak equivalence of simplicial sheaves" to mean the same thing as "induce weak equivalence on stalks", you must localize simplicial presheaves with respect to hypercovers.

You could only localize with respect to covers, but then it may be possible for there to be an object whose stalks are all weakly contractible, but which is not itself weakly equivalent to the terminal object.

Note: a "weak equivalence on stalks" amounts to the same thing as "isomorphism of homotopy-group sheaves", so I could have instead said: if you only localize with respect to covers, then it may be possible for there to be an object with trivial homotopy-group sheaves, but which is itself not weakly equivalent to the terminal object.

I highly recommend the paper of Dugger, Hollander, and Isaksen, Hypercovers and simplicial presheaves, which explains this, and gives an example exhibiting the phenomenon I just mentioned.

Added.

For "localize with respect to covers", I mean: from an open cover of an open set $X$ build a simplicial object $C_\bullet$ of presheaves, with $C_0=\coprod U_i$, $C_1=\coprod U_i\cap U_j$, etc. Then a simplicial presheaf $F$ has descent for covers if $$F(X) \to \mathrm{holim} C_\bullet$$ is a weak equivalence of simplicial sets (for all open covers $\{U_i\}$ of open sets $X$). Note that I'm taking a homotopy limit of a cosimplicial diagram here.

If you just use the "first two steps of the diagram", you would be saying that the map $F(X)$ to the "homotopy equalizer" of $F(C_0) \rightrightarrows F(C_1)$ is a weak equivalence. This is a different condition, and is the wrong way.

Here is one way to see why it is wrong (I'll be a bit imprecise here): start with the constant presheaf whose value is an Eilenberg-MacLane space $K(G,n)$, and "localize" it to get a simplicial presheaf $F$ which has descent for covers. I want to be able to say that $F(X)$ has something to do with the cohomology of $X$. With the correct definition (full simplicial diagram), you can see that you are seeing something that looks similar to the classical definition of Cech cohomology. In fact, if $\{U_i\}$ is a "good cover", so that all finite intersections are either empty or contractible, this is what you get. If you just use the "two-step" definition, it's not clear that $F(X)$ will know anything about 3-fold intersections of open sets in the cover, so presumably can't compute Cech cohomology.

This answer is not really different from David's or Marc's; I just want to boil things down.

In brief:

If you want "weak equivalence of simplicial sheaves" to mean the same thing as "induce weak equivalence on stalks", you must localize simplicial presheaves with respect to hypercovers.

You could only localize with respect to covers, but then it may be possible for there to be an object whose stalks are all weakly contractible, but which is not itself weakly equivalent to the terminal object.

Note: a "weak equivalence on stalks" amounts to the same thing as "isomorphism of homotopy-group sheaves", so I could have instead said: if you only localize with respect to covers, then it may be possible for there to be an object with trivial homotopy-group sheaves, but which is itself not weakly equivalent to the terminal object.

I highly recommend the paper of Dugger, Hollander, and Isaksen, Hypercovers and simplicial presheaves, which explains this, and gives an example exhibiting the phenomenon I just mentioned.

This answer is not really different from David's or Marc's; I just want to boil things down.

In brief:

If you want "weak equivalence of simplicial sheaves" to mean the same thing as "induce weak equivalence on stalks", you must localize simplicial presheaves with respect to hypercovers.

You could only localize with respect to covers, but then it may be possible for there to be an object whose stalks are all weakly contractible, but which is not itself weakly equivalent to the terminal object.

Note: a "weak equivalence on stalks" amounts to the same thing as "isomorphism of homotopy-group sheaves", so I could have instead said: if you only localize with respect to covers, then it may be possible for there to be an object with trivial homotopy-group sheaves, but which is itself not weakly equivalent to the terminal object.

I highly recommend the paper of Dugger, Hollander, and Isaksen, Hypercovers and simplicial presheaves, which explains this, and gives an example exhibiting the phenomenon I just mentioned.

Added.

For "localize with respect to covers", I mean: from an open cover of an open set $X$ build a simplicial object $C_\bullet$ of presheaves, with $C_0=\coprod U_i$, $C_1=\coprod U_i\cap U_j$, etc. Then a simplicial presheaf $F$ has descent for covers if $$F(X) \to \mathrm{holim} C_\bullet$$ is a weak equivalence of simplicial sets (for all open covers $\{U_i\}$ of open sets $X$). Note that I'm taking a homotopy limit of a cosimplicial diagram here.