If they don't much like each other, it might be best to provide them with separate cages so they can have their own private space, especially if there are signs of aggression between them. Living in a small space with a roommate you don't like can get really stressful for both birds and people. The cages can be kept side by side, if they seem to appreciate each other more at a slight distance. If they really hate each other, a further separation - like different rooms might be necessary, but it doesn't sound like it is quite that bad.
As far as the nest goes, the first question we are always going to ask when it comes to eggs is if you and your birds are ready to hatch them. Most people aren't really set-up to properly care for baby parrots and more problems will inevitably follow if the eggs are allowed to hatch.
In many cases, the best choice is to remove any fertile eggs as soon as they are discovered and replace them with dummy eggs or boil the eggs and return them to the nest (after marking them so you can tell them apart from any newly laid eggs later on). Keep checking the nest regularly for new eggs, until the female has stopped laying new ones. You can wait for her to abandon the nest on her own or remove the eggs after a week without any new eggs. After that, you can take steps to reduce nesting behavior by removing all nest boxes or "nest-like" toys, rearranging the cage, and adjusting diet or light levels to interrupt the breeding hormones. If all else fails, separating the male and female is a guaranteed way to prevent fertile eggs, but it is usually not necessary if you keep alert for nesting behavior so you can remove eggs quickly and take other measures to manage hormones.
If you are prepared to breed ... or if the eggs are close to hatching, so removing them from the nest is no longer a viable option, we are here to offer any advise and assistance that we can. Breeding parrots can be a deeply rewarding experience, but it is also quite a challenge, especially for first time parents. If these are your first eggs, do as much reading as you can now and make sure that you have hand-feeding supplies available before the eggs hatch! There won't be time afterwards, if it becomes necessary to step in and take over caring for the babies yourself.
Good luck and welcome!