All birds have their own different personalities, even those from the same species... So your bird will choose his or her own friends. In my opinion, it works better if the birds are close to the same size (there may be exceptions of course, but I would not put a large bird like an african grey with a small bird like a green cheek conure). Even birds close to the same size can be violent with one another, so do not automatically assume that you can bring another little conure home and they'll be friends.
That said, my 7-year-old female green cheek conure Lola lives with my 5-year-old male blue crowned conure Lincoln and my 2-year-old male yellow-sided green cheek conure Connor. They are all caged together and love each other's company- they get along very well. This was not always the case, though! Connor disliked Lola right off the bat and chose Lincoln as a buddy even though Lincoln is bigger than Connor. After a year of being around each other during play time outside of their cages, Connor began to really love Lola and I saw that when they were out of their cages Connor and Lola were cuddling and preening each other! I began putting all 3 in the cage together (it's a big cage, plenty of room for everyone) under *close* supervision and saw that they all got along very well. After several weeks of supervised in-the-cage time together, I let Lola move in with Connor and Lincoln. They have been caged together for about a year now, and it works well for them. They have plenty of perches, but I have noticed that all 3 like to share the same perch for sleeping and they all cuddle up together. I continue to keep a close eye on things so that I can separate them if problems arise, but they have not had problems so far. I guess it's like that show Sister Wives in a way, except that Lincoln and Connor are "Brother Husbands" to Lola.
By the way, my green cheek conure Lola was "married" to Karma the lovebird for 6 years until he passed away from old age. Every situation is different, so be prepared to cage birds separately until they can be completely trusted together.