I had the same issue, after chest day my shoulder was on fire for two to three days. I finally broke down and went to the doctor.
After some X-ray and Predisone shots, it was determiend that I had stage 3 arthritis in my joint between the collar bone and scappula,
I went ahead and had surgery done on it, they cut back the collar bone, to increase the distance, cleaned out all the torn up cartilage, and removed a bone spur.
The first work out I had was horrible, 8 weeks after surgery I could barely bench press the bar without it hurting.
I have a buddy that is an orthopedic surgeon, he recommend the following.
Warm up your shoulders really well, with a 5 pound weight, sit at a table and raise and lower the effected arm, similar to arm wrestling, this strengthens the smaller muscles in the shoulder joint responsible for stabilizing the joint.
He gave me a whole slew of joint exercises to do.
When doing flat bench presses, place a 25 pound plate under the foot of the bench, this slight decline, takes a lott of the stress off of the shoulder joint, also grip the bar in a suicide grip, ( Don’t wrap the thumbs) ths places the wrists and arms in the proper position.
When completing the actual press, don’t flare the elbow out and contentrate on lowering the weight to your nipples and then up in a “J” motion.
Alos hand position is key, too wide and you place the joint in a bind.
After a few weeks, the pain went away.
The surgery sucked, the re-coup sucked worse, but now I can press and don’t have any pain.