Despite the prevalence of unilateral breathing, some coaches recommend bilateral breathing to develop symmetrical body roll to each side and to avoid the lopsided stroke that often comes with same-side breathing. Michael Phelps breathes to one side, but in this training video, his coach, Bob Bowman, recommends learning bilateral breathing:

On CoachesInfo.com, aquatics scientist and masters swimmer Ross Sanders advises that same side breathing can interfere with streamlining:

... 'Twisting' of the upper body during breathing is common and increases resistance. Observation of swimmers indicates that this twisting is more common among swimmers who have a preferred breathing side. I believe swimmers should learn and practice bilateral breathing. Coaches should establish symmetry of action to improve balancing of rotations and streamlining.

Total Immersion founder Terry Laughlin is all about making your breathing motion so streamlined that it doesn't slow you down no matter how often you breathe, but even so he recommends bilateral swimming while training:

One of the most common questions I get from swimmers is whether they should use alternate-side, or bilateral, breathing. The quick answer is yes, you should breathe to both sides. At least in practice. And on some occasions it can be an advantage while racing too. 

... The problem with single side breathing is that, over time. it tends to make your stroke lopsided and asymmetrical. And small wonder; in just an hour of swimming, you'll probably roll to your breathing side about 1,000 times, meaning all your torso muscles pull more in that direction and less to the other side. Multiply that by hundreds of hours of swimming and you can see how a lopsided stroke can easily become permanent.

One would think that bilateral breathing should at least be attempted by any serious swimmer, but in an online interview with staunch bilateral breathing advocates at Swim Smooth, Swimming Fastest author and swim coach Ernest Maglischo cautions that teaching bilateral swimming might be a waste of time for some: 

I believe bi-lateral breathing is a good way to teach beginners because they will tend to be more rhythmic. But, I am of the opinion that competitors should breathe to only one side when racing. Oxygen consumption should be greater when more breaths are taken during the race. Having said that, swimmers in races should resort to breathing to both sides on occasion in order to check their direction and the position of their competitors. As for using bi-lateral breathing in the training of experienced swimmers, I have found that it is a waste of time. They will swim more symmetrically in training when breathing to both sides, however, they will revert to the same somewhat lopsided stroke when they breathe regularly in competition.

When I used to follow rec.sport.swimming, some posters claimed that asymmetrical stroking, or what they called loping, had less to do with breathing than with arm dominance, aka motor laterality. One doesn't see noticeable asymmetry in backstroke though, where one breathes facing up, or in breaststroke or butterfly, so I did not find such claims convincing. In 2005, Seifert, Chollet and Allard ran a study (PDF) that asked: 

... does an asymmetric arm pattern emerge from internal properties (functional pathology, dominance of one arm) or in response to external constraints (breathing)? And what is the direction of causality: Is the asymmetric pattern determined by unilateral breathing? Or, conversely, does an asymmetry due to arm dominance lead to unilateral breathing? 

[But their conclusion to the chicken or egg question was ... chicken and egg:]

This confirmed the relationship between unilateral breathing and coordination asymmetry, and suggests that coordination symmetry relates to both motor laterality and breathing laterality.

My bilateral breathing strategy for the last decade has been to breath to the right and left on alternate lengths. I breathe right going out and left coming back. I'm a 5% faster swimmer breathing to the right, though, and even faster when I breathe bilaterally once every three strokes. 

So despite some of the cautions, my experiment for this season has been to incorporate one right, one left (1R/1L) bilateral breathing into my long practice swims and all my sprint sets. To retrain my body, I quit same-side breathing altogether. I started by alternating 50m crawl lengths with 50m easy breathing backstroke lengths. On the first few swims, I was desperate for air before and after the flip turns, and gasped as I surfaced in backstroke.

But a few swims later, I was not feeling so bad after the turn, and stroked hard on the backstroke. By about 600m, bilateral breathing was feeling like the right and proper way to swim. Moving to the outside 25m lanes last Sunday, I swam three lengths of crawl for every one of backstroke. Due to a sprained ankle I was doing open turns, which gave me an extra breath, of course, and I felt no air desperation at all. I knew that flip turns would be more challenging.

One day I swam a full 1500m of crawl and 500m of backstroke. I was able to manage flip turns with tentative pushoffs. I found that extra breathing just before the turn tended to mess up my flip timing, and I had to tuck a lot to make the rotation. For about the first half of the swim I stuck with bilateral breathing--one breath every three strokes--but gave myself extra breaths before and after the turns. I even tried breathing on successive strokes, like Sun Yang, but I don't exhale fast enough to be ready for the next inhale. For a few lengths I tried breathing twice to the right and once to the left (2R/1L), and eventually settled into breathing twice to the right and twice to the left (2R/2L), which seemed to be enough air. Both 2R/1L and 2R/2L average four breaths per ten strokes, but the rhythm differs. 

After breathing 2R/2L for several weeks the rhythm became routine enough that I could take time to count strokes. I felt like I was taking a lot of strokes, but my Poolmate swim watch had been confusing me. It counts arm strokes on the watch hand to give an average stroke cycle count per length, which should be fairly consistent between swims. But I saw 14, then 7, then 6 in the 25m lane, and 32, then 15 in 50m lane. I was convinced that the watch was going bonkers until I realized the problem.

When I finished a long swim, I was sometimes pushing the start button too briefly, then taking off my goggles, looking at the results, etc. That meant the timer was paused rather than stopped, so the watch would start another session. That second session would only end when I pressed again to switch back to Clock mode. Since there were no strokes in the second session, the watch gave me an average of both sessions (14 + 0) /2 = 7. Or (31 + 0) /2 = 15. 

14 left arm strokes should mean 27 or 28 arm strokes, and 11 breaths, but I actually count 25 strokes, so the watch probably interprets balancing arm movements during the flip turn as a stroke or two. Last season I started at 25 and got down to about 18, so now I'm in the Ministry of Silly Swims until I can lengthen out my strokes again.

For now, the 2R/2L pattern is giving me enough air for longer swims. And when I do sets of shorter crawl swims, the 1R/1L pattern feels very smooth and natural. And I have the rest of the summer to see how it works out.