I was in a very successful partnership with my best buddy for nearly 20 years. My thoughts...
Limit the number of partners. The first partner saves each partner 50% of the fixed costs. A third partner only saves an incremental 17% more. A fourth only 8% more. Like JohnH said, two people can usually work things out... with three, there is the possibility that one will be the odd man out. I don't even want to think of 4+.
Get a partner with similar flying needs. Over our 20 year partnership deal, our flying hours were pretty much the same every year. The way we worked it was every other week was someone's priority week. If it was your week, the plane was there for your use unless you allowed your partner to use it. Over 20 years, we only had I think three or four times that we both really wanted to use the aircraft at the same time.
Get a partner with similar economic means. We were able to agree on many upgrades, and flying was not a strain on our finances. I've seen partnerships fail simply because one partner could not realistically afford what the other partner wanted.
Get a partner with a similar flying and safety attitude. I knew exactly how I would find the aircraft if my partner flew the aircraft before me. He was meticulous and safe, and didn't take risks. We got along just fine.
Split the ownership tasks evenly with your partner. In our case, my partner took care of the books, insurance and hangar, I took care of the maintenance.
Settle up on a regular schedule. We did ours annually... and rarely did one owe the other more than 1 AMU.
Fly together often. Whether it be for currency or $200 hamburgers, good partnerships are about friendship as much as flying. Frequent touch points also limit any brewing concerns.
Know that the breakup is coming someday. In our case, my partner retired and moved to Colorado. I bought his share... in fact the "price" discussion lasted less than five minutes.
In reality, I miss having my former partner around..... miss the friendship and comradery, the flying... and somewhat the 50% of additional fixed costs I am now paying.
If you can't afford the aircraft that you really want, IMO adding a partner is a good way to have the best of both worlds.
David
P.S. despite the conventional wisdom, we never had a written contract for the partnership. We both came at it from the perspective of mutual respect, and a handshake deal meant something. Flame away if you must, but I don't regret this at all.