I had a right shoulder dislocation around three months ago. I am back to weight training now, and the shoulder feels great, but for whatever reason, my LEFT side is now weaker in almost everything I do. Pull ups (I could previously do about 30) now are completely awkward because my left side has no strength, and I am using an assisted machine! Bicep curls bring up a huge strength difference as well.
There are few reasons you may be experiencing this loss of weakness on the left hand side after you had dislocated the right shoulder. The first reason is that whenever you injure one side, the other side also gets affected. It depends on the mechanism of injury.
If it was a severe enough injury to the shoulder joint that it dislocated, the force to dislocate it would affect the other scapula and shoulder joint. That would cause proprioception alteration throughout the shoulder girdle and A/C joints and therefore possible weakness.
Another reason is that the cervical spine may be affected from the original trauma, thus causing an inhibition of the brachial plexus that innervates or gives power to the shoulder and bicep muscles.
The third reason is that the upper thoracic spine may also have been affected, causing what is known as "T4 Syndrome" and also causing a weakness in the left arm.
The fourth reason is a common reason but rarely discussed. It’s that you may have injured the left side independently of the right side. We sometimes cause an injury, and we don’t know it. An example would be falling asleep on the couch watching TV with your head in an abnormal position and your arm outstretched over your head. That could cause a traction injury on the nerve enough to alter the neurological firing and thus decrease the power output of that arm. We tend to try and relate one injury to another, and sometimes they relate and sometimes they happen independently of each other.
So please get a proper assessment of both shoulder joints, the cervical and thoracic spine, and the proper treatment and rehabilitation will get you back to doing 30 pull ups in no time at all.