The persistent skill system makes playing as a human much more interesting. There should be a skill tree for zombies as well, which would make playing as a zombie more enjoyable.
I would suggest that the two trees be separate, but each has the same amount of points available to it. A player with 5 skill points puts 5 points into the human tree however they want, and 5 points into the zombie tree however they want. When they level up, they get an extra point for human and an extra point for zombie.
In the same way the human tree works, skills would offer trade-offs, or, the skills would be only situationally useful. Some examples off the top of my head:
-Increased movement speed, but decreased damage to humans (For attacking barricades or chasing humans)
-Increase damage to props, but decrease damage to humans
-Slow humans you hit, but decrease attack speed
-Reduce damage taken, but slow movement speed
-Increase the resist-o-meter for other nearby zombies, but lower it for yourself
-Deal much higher damage to sanity sigils
-Increased speed when no humans are nearby
-Regenerate health much faster when feigning death
Zombies can pretty much have the same sorts of powers as humans, although since the different zombies have such different movement speeds, health pools and attack rates, things should probably be based on percentages instead of flat values. (+4% health instead of +4 health, for example)
But like I said, these examples are just off the top of my head and may not be balanced. The point is that having a skill tree for zombies would make playing as a zombie more entertaining by allowing you more control over how you want your zombie to function.