I am a side sleeper usually. Can I sleep on my side after a BBL if there is no fat transfer to my hips?

After undergoing a BBL, if there was no fat grafting to the hips - then sleeping on your sides should not be an issue. (Make sure with your surgeon before doing though.)
The idea is to avoid excess pressure to any area that has been grafted with fat - to allow for maximal fat survival. In my practice - I tell patients no sleeping or sitting on anywhere I grafted for minimum of 3-4 weeks. 
Hope this helps. 

Dear Makingmybodygreat2018,I instruct my patients not to sit for 2 solid weeks. We provide them a specialized pillow that our patients can sit on during that time. They can sleep on their stomachs or sides. Daniel Barrett, MD, MHA, MS 
Certified, American Board of Plastic Surgery  
Member, Am. Society of Plastic Surgery

I normally recommend patients to sleep on the stomach since fat is transferred to the buttocks and hips. Consult your surgeon for post op recommendations.

Hi and Happy Holidays. You can sleep as you wish, side, on your back, etc. Its a myth that the fat will  migrate. Member of the American Society of Plastic Surgeons (ASPS)
Member of the International Society of Aesthetic Plastic Surgey (ISAPS)
Certified Plastic Surgeon 25 years of experience

I know it is hard, but I would recommend waiting for 2-3 weeks before sleeping on the sides. Since most of us graft the sides of the buttocks to increase the fullness and give a nice rounded look, I would suggest you spend as little as possible on all the grafted areas to maximize fat cell survival in the initial period when blood vessels penetrate the grafts.We want to maximize the flow of oxygen into the grafted areas and prevent compression. Ideally we want to maximize a period of no compression on the grafted buttock areas  for 2-3 weeks. 
I hope that you find this answer helpful & Good Luck on your recovery!