If you have a homomorphism to $K'$ where $K'$ acts on a set $X$, then you could use ping-pong arguments to verify injectivity. In your setting, this is called 1-st Maskit combination theorem, you can find some examples in Maskit's book "Kleinian groups". If $K=G_1\star G_2$, then ping-poing setting is a pair of subsets $A_1, A_2\subset X$ so that $A_1\cap A_2\ne \emptyset$, and $\forall g\in G_i\setminus 1$, $gA_i\subset A_i^c\cap A_{i+1}$, where $A_i^c=X\setminus A_i$, $i=1,2$ mod 2. Under this condition, you do get a faithful action of $K$ on $X$.
Similar thing works for more general amalgams $K=G_1 \star_H G_2$, but the ping-pong condition becomes more technical.
Edit: Here is the ping-pong setting for general amalgams (Maskit's 1st combination theorem). You have an action of a group $G=G_1\star_H G_2$ on a set $X$, where each $G_i$ acts faithfully and you are trying to determine if the action of $G$ is faithful. A subset $X'\subset X$ (which need not be $G$-invariant) is partitioned in three (disjoint, nonempty) subsets $U_1, U_2, S$. You should think of two complementary components ("open half-spaces" or "halfs of the ping-pong table") to a hypersurface $S$ ("the net on the ping-pong table") in a manifold $X$. If $X$ is a complex manifold, you are in trouble since such $X$ does not come equipped with natural real hypersurfaces. One frequently uses a bisector for a suitably chosen metric on $X$ to define such $S$. Define $V_i=U_i\cup S$, $i=1,2$ ("closed half-spaces").
The ping-pong assumption then is that each $V_i$ ($i=1,2$) is strictly invariant under the subgroup $H\le G_i$, which means that $h(V_i)=V_i$ for every $h\in H$, and $g(V_i)\subset U_{3-i}, \forall g\in G_i\setminus H$. (The terminology strictly invariant is Maskit's. Notice that $U$'s and $V$'s both appear in the definition.) Examples could be found in Maskit's book.
If you think of $g\in G_i\setminus H$ as a player on the $i$-th team, then it sends the ball
(point in $V_i$) from its side of the table (including the net) to the opponent's side $U_{3-i}$.
If such partition of $X$ exists, then the action of $G$ on $X$ is faithful. The proof is a direct application of the normal form for the elements of $G$. Namely, write $g\in G$ as the product, say, $g=h_0 g_1 h_1 g_2 h_2 ... g_{n}$ where each $h_j\in H$, $g_{2m+1}\in G_1\setminus H, g_{2m}\in G_2\setminus H$. Now, pick a point $x\in S$ and inductively apply each letter in this product to $x$ and check where the image lands: You will see that it never returns to $S$, unless $g=h_0\in H$.
There is a similar combination theorem for HNN extensions ("2nd combination theorem").
