MacBook – Headphones only play right side

It is probably some dirt in- or another problem with your sound-output jack.

If twisting your headphone-cable or repeatedly plugging it in and out of your MacBook does not work, you might fix it by doing something like (gently!) poking into it with the backside of a matchstick or trying to clean it with some cotton buds. In that case make sure you have one that does not fall apart easily or you'll end up with a cotton-filled connector.

If that does not work, I'm not sure if you are able to fix it yourself since your MB might need to be taken apart to get into the jack and clean it…

Depending on the breakage, you may not be able to control the software if the "jack detection" sensor is intermittent. It will over rule whatever software setting you have made and send output to the headphone jack each time it senses an insertion.

Often taking this to Apple at a genius bar is no cost if they can dislodge the broken piece with experience (and same for reputable repair shops). Also, you might only pay labor if the jack isn't broken. Usually there is more damage since why would a protected piece break off inside a perfectly functioning receptacle, but it's possible you broke the jack mostly or entirely before inserting it.

Give a shot at setting the sound in the System Preferences, but most of the times this happens, you'll need to get a repair if you can't use software to switch the output once and have it stick.