Net Force on Individual Masses, or the entire system? In the picture, would the net force on each individual mass take into account both masses or just the individual mass? For example, would the net force on the first mass be equal to $am_1 = F_{\text{net}}$ or would it be $a(m_1 + m_2)  = F_\text{net}$? 

 A: I wrote a whole answer, which follows below, but I want to preface by saying this:  When you're solving net force problems, it's very helpful to identify your system.  If you only want to know the net force and/or acceleration of block 1, then your system is just block 1.  In one of my intro physics classes, I was even taught to draw a loop around my system in the diagram, just as a double check.  Once you have identified your system (e.g. the objects you choose to solve for) then that is the mass you want to be using in your net force equation.
When you're looking at the net force equation for single object, you want to only use the mass of that particular in your net force equation.  You can convince yourself of this by setting up a free body diagram for each object.  So in this situation, you would want to use $am_1=F_{net}$. However, the second equation that you wrote, $a(m_1+m_2)=F_{net}$, would work perfectly fine if you wanted to solve for the net force acting on the entire two-block system.  I wouldn't recommend that approach on a problem like this, but it is a valid approach in other situations.
EDIT:  In response to some comments on another answer, it is worth pointing out that if you use the second method, $F_{net}=(m_1+m_2)a$ to find the acceleration of the system (in this problem, or any other similar two-body problem), it will only give you the magnitude of the acceleration, because block one and block to are moving along different axes.  In order to find the direction of acceleration for each block, you would have to inspect your free-body diagrams for each block after solving for the magnitude.
