I'm scared of the aggressive behavior my dogs show to each other in my multi-pet household
This is a common problem for multi-dog households.
The canine society is absolutely dependent on their social hierarchy. Sometimes you will luck out and have a bunch of mid-pack dogs in one family, which makes it very easy for the humans to deal with, but they will even work out a rank system.
Having an Alpha dog doesn't mean an end to multiple pets or family peacefulness, but it may require a little intervention on your part to keep the peace.
Often times, at our rescue, we do not notice a particular difficulty because the law is laid down from the very beginning and the new dogs just have to fit in. However, we also are not terribly disturbed by dogs growling, snarling, or even snapping at each other as this is their way of working things out. This can be expected for weeks as they get used to established sleep, nap, toy, food, etc... areas and claims.
The alpha will continue to occasionally assert themselves to clearly stake their claim as the leader. Believe it or not, this is reassuring to the other dogs. They cower, roll over, lick the chins of the alpha as respect and submission, not necessarily as a personal demonstration of anxiety or fear. The pack leader establishing themselves is actually reassuring to the other members as they are safe and protected and that they do not need to sweat the small stuff since that is the leader's job.
Having said that, most families do not enjoy continued strife in the home. In general, it is best that the humans establish themselves as the leaders of the home. This can bring the greatest peace. The dogs will have their own hierarchy, but the humans should still be at the top.
If two dogs get too rough or the alpha is being too rough or aggressive, they BOTH get reprimanded and put into time out (in a crate, in a room, etc...).
You absolutely cannot favor the non-alpha, this will actually make it much worse for him/her. You must establish what you say goes and that while they may squabble, when you say enough, that IS enough.
Taking them to an obedience class together (two family members) and working them side by side regularly will reinforce this family order. Making them perform tasks side by side. Each sit on a side of you and stay for a cookie and neither get the treat without both behaving.
If they quarrel, command them to sit/down/stay in the same room, then if they behave on release, yeah! If they don't, reprimand and crate them both.
Dogs are a lot like toddlers and you can train them very successfully using good teaching techniques and create a happy home. Being an alpha is a good thing and is a personality you cannot "get rid of", but they are often the most rewarding dogs you can own and certainly make for a safe household.
A little training time, being extremely consistent and calm, and establishing clear expectations, will create a happy home. If they snap, time out. If they fight (dogs in the same home rarely do significant damage) get a broom and time them both out. Do not get yourself hurt. Catch them at the first growl and snap at them and put them in time outs. Do not get panicky and worst of all, never fear your own pet.
Train your pets together, have high expectations and hold the dogs to them. Assert yourself as the one who delegates rank and home peace and command their respect. When they start growling or lipping off to each other (the equivalent to "mom, he's on my side. mom, he's touching me.") Handle the situation then. Time them out and make them do some obedience tasks to further assert you are the leader and they can cut it out.
Expect there will be quarrels though, just like kids. They will fight over a toy, a bed, etc... We don't kill our kids for fighting with their siblings. Dogs just make more noise and have teeth instead of opposable thumbs.
It is impossible to set up an individualized plan without meeting dogs personally, but I hope that this general information gives some insight and can help.
Written February 2007 by Elizabeth Riddle, President of the Central IL GSD Rescue.