Typically speaking, although it would be considered unusual if dodrio heads didn’t bicker amongst themselves, fights rarely turn violent. The reason why is because although they may not see eye-to-eye (forgive the pun), they each understand that they share a body. It’s even speculated that any damage done to one head is actually felt by the other two. For that reason, violence between heads (aside from light, annoyed pecking now and then, which you should fully expect and have probably already watched yourself) is extremely rare and would indicate a far larger, ongoing problem that you likely would have detected well before the dodrio grew to be of breeding age.
In a similar fashion, the funny thing about dodrio is that although each head possesses a separate brain, all three of them understand their body in terms of a unit. So while they may hold separate opinions, they will always do what’s best for them as a whole—and, for that matter, they will recognize possessions of theirs as theirs, rather than something belonging to a single head. This includes eggs: dodrio see their own eggs as a product of the three of them as a whole—which, to be fair, is completely true, as dodrio only have one reproductive system.
So, in short, you won’t have to worry about the heads fighting to the point where it becomes a real problem. At most, they may annoy each other to the extreme, but because they share a body, a dodrio head putting forth active efforts to do serious damage to one or both of its siblings is nearly unheard of—except, again, in cases in which the dodrio has extreme mental health issues and needs a comprehensive therapy program for stabilization.
Now, all three dodrio heads going up against the other dodrio is an entirely different matter, but given the aggressiveness of the typical dodrio, this may be a problem that may be difficult to correct until the hatchling has emerged and grown enough to be independent. Just … just make sure the heads don’t get tangled up with each other, and you’ll be fine. Possibly.
Come to think of it, be sure to wear a sturdy mask and thick gloves when you untangle them too.