What factors govern the placement of a helicopter's tail rotor (left or right)?
It's hard to say the design rational used for all helicopter manufacturers. Push vs pull doesn't seem to be very important. 

One example is the Bell UH-1 Huey (Left side Pull). When Bell modified the design for civilian use 205, 212, 412, 214, they put the TR on the (RH side Push). When asked about the reasoning for the original Huey/205 change, Bell engineers explained that they found better TR effectiveness with rotation of forward blade climbing, as the interaction with wake and vortices of the main rotor was more favorable in this configuration. 

Now look at the AH-1 Cobra Gunship variants. Bell switched sides at least twice. Left on early models, switched to right on later 2 blade models (AH-1W). Back to Left side for the 4 blade AH-1Z. While on opposite sides, the AH-1W and AH-1Z are both forward blade climbing. 

Sikorsky? S-76(Left - Climbing) H-60(Right - Climbing) H-53(Left - Climbing). One could speculate that the higher the TR is mounted, the more this rational comes into play.