I've had surgery on both rotator cuffs, due to injuries suffered playing basketball.
Not necessarily advice, but my experience:
I started training with bands, then clubbells and kettlebell DLs, on my uninjured side just as soon as I could without negatively involving the surgical side.
Then, once I was out of the sling, but still recovering and unable to do ballistics or presses with the surgical side, I did a few months of pretty intensive training, mainly A+A snatches, on only the good side. I basically did the same volume I would normally do, but with only one arm instead of split between arms. I made a few adjustments to rest periods and sometimes overall volume to account for increased grip fatigue, but didn't dial back the volume much overall.
My injured shoulder was my non-dominant side, so I was doing all the training on the side that was already stronger.
There were absolutely no negative consequences of doing this. During my recovery, the surgical side had a lot of atrophy, but once I was cleared to train normally it quickly regained size and strength and caught up to my dominant side (within my normal range of difference between dominant and non-dominant). I strongly believe that the unilateral training actually helped the recovery of the injured side.
In consultation with my doctor and physical therapists, I took the attitude that anything that didn't involve my injured side in a way that might interfere with recovery was fair game.
BTW, a few post-surgery essentials:
--A comfortable upholstered chair to sleep in. I needed to sleep in a chair for many weeks before I could sleep lying down.
--A good sling. If you have to live in an arm sling for weeks and months on end, even small differences in comfort add up to big differences in tolerability over time. Having your arm immobilized is just a very uncomfortable feeling on it's own, and there is pressure on the neck/opposite shoulder from the strap/harness.
In my experience, essential features of a tolerable sling are an abduction pillow and an offloading harness with a second underarm strap to pull the main strap away from your neck, direct the load more vertically onto the uninjured shoulder, and distribute some of the load under the arm of the uninjured side. It does take some experimentation to get the lengths of the different straps right for a comfortable fit, but it's definitely worth the trouble.
Some slings with this design are the Donjoy Ultrasling Pro, Ossur Formfit, and Breg Slingshot 3. Your doctor will probably send you home with a random sling that may lack an abduction pillow and/or offloading harness. It is well worth it to get the best sling you can, and get it ahead of time. For my second surgery, I got several different models to try because I knew how important it was, and ended up rotating a few different ones, since they each had different aspects that made them more or less comfortable, and even mixing and matching straps and other parts to get the best combinations.
--A good ice system. I'm not convinced that ice helps healing, but it definitely helps a lot with pain. A good basic cold pack that works well for the shoulder is the T-shaped Cryomax Pro. It gets very cold, stays cold a long time, and the T-shaped model fits well over the shoulder. It's by far the best basic cold pack I've used. Get at least two so you can rotate them. Then there are ice buckets that pump cold water through a cuff that straps around the shoulder, such as the Aircast Cryocuff. And then there are units that refrigerate the cold water themselves, so you don't have to put ice in them, which are the most expensive. My doctor set me up with one of the fancy refrigeration units, that my insurance paid to lease while I needed it, but I found out that they were charging my insurance thousands and thousands of dollars for it, whereas an Aircast or similar system is a couple of hundred dollars and multiple cold packs are less than a hundred. So even though I wasn't paying out of pocket for the refrigeration unit, I felt bad about using it.
