I had to fix my rear passenger side a little while ago. When you pull on the door hanle your actually actuating a rod that pulls on the mechanism that opens the door. The rod is attached via a mechanism that has a screwed adjustment.
It could be out of adjustment, or it could need to be replaced.

Unfortunalely you have to remove the door panel to get at it. It is not a difficult job, just proceed with care so that you dont damage any of the little parts that hold the door panel and various fittings.

I found the parts that were required, in abundance, at the wrecker.

Word to the wise: after you get the work done, dont be in a hurry to put the panel back on. Be sure everything is working properly. Take your time.
I had to buy another panel($45). The door would not open. I had to destroy the panel in order to open the door.