Depends sooo much on the individual dogs. If you are going the rescue route, see if you can do a few meet & greets or even a trial period to find a pup who gets along nicely with your current one. We have a spayed female (aussie/lab), a neutered male (boxer mix), and an intact male (cavalier), and all get along wonderfully...never any scraps between them. BUT, all them individually are very docile, mild-mannered dogs. So none of them are really interested in challenging each other.
My parents on the other hand, had two neutered males who fought very badly (one would have probably been killed if they didn't break them up) needless to say, they re-homed the younger dog to my brother. They both have scars from breaking up those fights. BUT, both of those dogs have issues with other dogs in general...not just with each other. Both also have resource and space guarding issues. So they definitely didn't mesh well together.