Why breath bilateral you may ask?
There are a number of reasons why being able to breathe bilateral is beneficial.
The first one is to improve your balance and symmetry in the water. When you watch world class swimmers racing and they are only breathing to one side you may say 'well if it is good enough for them why not me'. You need to understand that the majority of these swimmers will be doing at least 80% of their training bilaterally to improve their balance and symmetry in the water.
Another area where being able to bilateral breathing is beneficial is in open water swims. Being able to comfortably bilateral breath can reduce the distance you are swimming as you are more symmetrical and balanced in the water and less likely to go off course. It also allows the ability to handle choppy conditions better where you can breathe away from the waves or choppy conditions or if you want to keep an eye on the person directly to your side that you are drafting off. You may end up swapping from one side to the other depending on conditions but if you are just a unilateral swimmer you won't have this ability.
If you are having shoulder issues and are a unilateral breather it may be due to poor rotation on your non breathing side. On your breathing side you will likely have good rotation, maybe even a bit of over rotation leading to a scissor kick but on the opposite side you may have developed a flat recovery arm across the top of the water. By developing bilateral breathing you develop better rotation to the non dominant side and therefore become a better all round swimmer.
During a race you may predominantly breath to one side but if you are comfortably bilateral breathing and do most of your training bilateral the balance and symmetry developed in your stroke will still be there.