It will vary from privy to privy, so follow the instructions that are provided at the privy.  The moldering process requires some moisture, but too much moisture can impede the process, and depending on the design, amount of use, and current conditions, could cause unhealthy, unsightly runoff and unpleasant odor.  Use your brains and consider the situation.

In the absence of other guidelines for the specific privy/conditions, here are my recommendations: 

Guys, unless it would be a public embarrassment (think of others more than yourself,) pee in the woods.  Why wouldn't you?  It's much more convenient, and avoids adding to any "too moist" issues in the privy.  In addition, you avoid the possibility of making the seat area, um, er, "unpleasant" for the next person to use.

Gals - It's much more of a hassle to go in the woods.  You're welcome to use the privy.