Now let's not jump gun. Weakness in the shoulder can come from different things such bad coordination, problems with the nerves, muscles or tendons.

If it is caused by bad coordination, then stretching and proper strength training should help, along with more attention to punching technique.

If the surrounding muscles and nerves are fine then it may be joint related, say the rotator cuff which is more common, which also could be caused by weaker back.

Bad punching technique can cause problems, with the straight punches, don’t raise your elbows before you punch because that would cause more strain on your shoulder joints. A bad stance such as raising your shoulders or hands higher than natural can cause muscle soreness, and fatigue, especially when it comes to punching and keeping your hands up.

Overtraining can also cause problems say for instance your are serious weightlifter who’s also undertaking a boxer’s regimen a few days a week, happened to a good friend of mine, it destroyed his shoulder, now he can’t train for boxing anymore.

If it’s simply a weak shoulder muscle than it’s pretty straight forward, if train your shoulders and the surrounding muscles (chest and back) the symptoms will be gone.

Correct me if I’m wrong, you’ve never had right shoulder banged up, or an infection in your shoulder. You don’t feel stiff or tight in your shoulder, back or chest. If not then great then we can move on to what you can do.
Let’s cover all the bases.

Step 1. Stretching:

A couple good ones, front crossover pulls. You’ve probably seen this one before, what you gotta do is put your right arm behind your head and with your left hand, grab your right arm above the elbow and pull it to your left. Hold each stretch for about 10 seconds and repeat 3-5 times.

The next one is squeezing your shoulder blades together. Start by bending your elbows together with your forearms sticking it infront of your body, push your elbows back to squeeze your shoulder blades together. Hold for 10 seconds and repeat 3-5 times.

There’s other stretches that’ll help too. What you want is to loosen up your chest, upper back and shoulder muscles, so they won’t have any more trouble working together.

Step 2. Boxing Technique.
Stance: Make sure that your hands are not at head level unless it’s an emergency. Keep your shoulders relaxed, and don’t raise them or tuck your chin into the side of your shoulder unless it’s emergency. That should help your shoulders but if you want a detailed explanation on proper stance take a look at the expertise of “greynotsoold” his example is literally straight out of the textbook. http://www.saddoboxing.com/boxing_fo...-t30092.0.html
Also make sure you don't raise your elbow early this will literally destroy your shoulders.

As for strength training I think there’s a couple good free-weight exercises that should help strengthen your shoulder. Look them up lateral raises, isometrics/static- holds (free-weights). Make sure your work is progressive in which you are building up range of motion, strength and endurance.

Wow did I write that much? If you’re reading this far you must be 110% serious. If that doesn’t resolve it be a rotator cuff problem, joint, or something else entirely. I couldn’t tell you then until you know what the cause is. Doctors will have better odds taking you through hoops of tests of finding out what it could be. You can also get a second opinion.