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](http://www.math.uiuc.edu/K-theory/0563/), 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.