Will bursitis travel from one shoulder to the other?

No. The only way bursitis would occur in another area is if you were doing an activity on the other side that would also cause it. For example, many golfers get it on both sides because they do use both hands to swing the club.

it shouldn't. Bursitis is an inflammation of your bursa, usually caused by carrying a purse or backpack that's too heavy. it won't move from shoulder to shoulder unless you switched the really-heavy whatever to the other shoulder.

Only if you do the same activities which inflamed the bursa of the shoulder in pain now. Bursitis doesn't travel....it is an inflamation or the bursa which surrounds the shoulder joint.

Here is a more technical explanation:
Impingement syndrome, or shoulder bursitis, occurs when there is inflammation between the top of the humerus (arm bone) and the acromion (tip of the shoulder). Between these bones lies the tendons of the rotator cuff, and the bursa that protects these tendons.

Normally, these tendons slide effortlessly within this space. In some people this space becomes too narrow for normal motion, and the tendons and bursa become inflamed. Inflammation leads to thickening of the tendons and bursa, and contributes to the loss of space in this location. Eventually, this space becomes too narrow to accommodate the tendons and the bursa, and every time these structures move between the bones they are pinched--this is the impingement.

What causes this problem?
Impingement syndrome is a descriptive term of pinching of the tendons and bursa of the rotator cuff between bones. In many individuals with this problem, the shape of their bones is such that they have less space than most others. Therefore, small thickenings of the tendons or bursa can cause symptoms.

Often there is an initial injury that sets off the process of inflammation. Thereafter, the problem can be self-exacerbating. Once there is an initial injury, the tendons and bursa become inflamed. This inflammation causes a thickening of these structures. The thickening then takes up more space, and therefore the tendons and bursa become are pinched upon even more. This causes more inflammation, and more thickening of the tendons and bursa, and so on.

What are the symptoms of shoulder bursitis/impingement syndrome?
Common symptoms include:

Pain with overhead activities (arm above head height)
Pain while sleeping at night
Pain over the outside of the shoulder/upper arm
Impingement syndrome and a rotator cuff tear are different problems, and although they are related, the treatment is different. Impingement syndrome is a problem of inflammation around the rotator cuff tendons. A rotator cuff tear is an actual tear within the tendons. The best signs that differentiate these problems area the strength of the rotator cuff muscles. Your orthopedic doctor will be able to specifically isolate these muscles to better determine if a rotator cuff tear is present. A MRI can also show the tendons of the rotator cuff and help determine if a tear is present.

While looking for your answer, I stumbled upon this site...Very informative...
http://www.headandneck.com/book/Chapter5...