Pushing, shouting, "boxing" like rats (standing on hind legs and slapping) and going around each other is fine and normal dominance stuff. If it gets really rough, like they're going at it for hours, kicking each other really hard, actually biting or otherwise obviously trying to harm the other and remove him from the area, then that's more serious.
The most serious fight situation is the "Furball of death" (often called the FBOD), which is when a degu actually grabs another degu, full on bodily grabs them, and attempts to start biting vulnerable parts. This will result in the other degu doing the same, and they become a ball of fur that will result in someone being seriously injured or dying unless stopped. Basically, when someone attempts to grab hold of someone else so they can start biting, you know things are really serious and you must immediately, in seconds, break it up.
The rule of thumb is that physical contact, pushing, shouting and slapping is alright, but any actual attempts at physical harm are not. If they attempt to start biting or anything like that, break it up immediately.
For this reason, it is very important to do introductions in a place where they cannot get into some sort of a nook or cranny where you cannot reach. You must be close to them constantly, and observe their actions very readily. If it looks like someone is going to grab someone or if you see them chasing each other trying to bite, break it up immediately. To break it up, there are a couple of methods. You should have a thick glove on when you are unsure about how they will react (you won't have time to go grab it if things look bad, and you do NOT want to try to put a bare hand in the middle of that). If possible, grab someone and physically take them away from the situation. The most important thing to breaking it up is to prevent physical contact.
If you need to break up a fight, use anything you can to first break contact. A glove, a blanket, I've even just used my phone to quickly push one away from the other so that I can get more stuff in between them. The last time, I threw a blanket over one of them after pushing them apart. Then I quickly separated them more.
If a serious fight situation happens, you will need to roll your introductions back and let things settle down. Return them to separate cages, allow them a day or two to calm down, and start the whole process over, from having cages next to each other. Inspect how they're behaving. If they're still furious or aggressive, give them some space and more time to calm down. Basically, start the process again and test how far you can take the basic steps. It can be a very long process.
And I really must stress: wear a thick glove. When rodents are in fight mode, they will NOT differentiate between the other degu and your hand. The most painful injuries in my life were caused by not respecting this.
A large sieve is good for containing an angry degu once you have broken up the fight - they will try to restart the affray so you'll need to secure one before removing the other, just make sure you watch out for trapped tails etc.
Low level aggression is normal, and its usually noisy and vocal, if its quiet fighting its a bit more serious.