My best guess would be that you have glazed rollers on that weak side or you have some gearing issues.
Using a flashlight, you might be able to see the rollers in action as the machine tries to print. If no rollers are turning on the weak side, then it's likely you have a gear problem.
With the front open as if you were going to change ink tanks, take a look around at the gearing for something that might look amiss.
Sometimes dust and/or bits of paper can cause this, esp. if it's a top-feed.
A blast of canned air wouldn't hurt.
If the machine looks OK mechanically, then glazed rollers would be the likely culprit.
There are products such as the following that clean and restore a printer's rollers.
In a pinch, no pun intended (please work safely), one could substitute Windex w/Ammonia-D (or 90+% Isopropyl Alcohol) and a paper coffee filter to scrub the rollers.
A coffee filter is relatively lint-free as opposed to a paper towel or Q-Tip and you will be able to scrub harder with it.
Windex will cut most inkjet ink but it won't rejuvenate rubber like the commercial products will.