Is it that likely for my chin implant to shift in my sleep the first couple days/nights after surgery?

It would be very unlikely that sleeping on your side would cause your chin implant to shift in the first couple of nights after your surgery.  It is also not unusual to have more swelling on one side versus the other.  I would not be concerned at this point and it is likely that things will even out as the swelling subsides.  

I would not worry at this point. The pocket for implants is created just large enough to accommodate the implant so there is not much chance for migration. Asymmetrical swelling is common and occurs regularly with this procedure. In 4-6 months, if you have any concerns about the aesthetics, follow up with your surgeon and discuss your concerns. Your sleeping patterns should not influence the position of your new implant. Good luck

It is unlikely that you have shifted your implant, but if you are concerned, I recommend following up with your surgeon. Swelling is typical and can be asymmetrical. It will gradually resolve over the next several months. I hope this helps!