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By paying attention to both your budgies' behaviours, you will be able to tell if they are coping well with the separation or not.
From your description, it seems your budgies are doing quite well while separated. They are happy, chirpy and able to lead their normal lives without a care in the world.
If they show no signs of being affected by this change and don't seem stressed or anxious, then you can continue with this separate housing arrangement.
Since you mentioned your budgies argue quite a bit, maybe they are truly better off and happier by being housed on separate cages.
Whenever there is consistent, daily bullying, where the aggressive budgie will chase the mate, keep him/her from eating well, playing with a given toy or standing on a preferred perch, immediate separation is needed. Immediate and permanent separation also applies when there is feather plucking during a fight and when blood is drawn.
Given your male's more shy and reserved personality and the fact that he is an adult budgie, it will take more time for him to open up and be more receptive to you.
Still it's possible for you to form a bond with him and this begins by building the foundations of trust with your budgie, to connect with him by using your voice and make him realize that you are not a threat to him.
This can be achieved by spending daily quality time with Sky by sitting close to his cage and talking to him in a calm, positive, encouraging and reassuring way.
With time he will start to be more receptive to your interactions, he will feel perfectly at ease around you and will even move closer to you while in the cage.
When this happens, you can start to offer him a tasty treat through the cage's bars and later on, you can place your hand inside the cage to give him the treat and encourage him to step onto your hand/finger.
By looking at both your budgies' body language, respecting the safe boundaries set by them and to proceed with the next step in training when your budgies are ready to do so, you will be well on your way to having them tamed and trusting of you.
The level of tameness can vary depending on the type of bond you have with your budgie and the budgie's overall personality.
Some budgies would at the very least only be finger tamed while others can be fully tamed.
For all the detailed information and tips on how to tame your budgies, you can check the sticky threads at the top of Taming and Bonding main page.
Good luck with Sky and Willow!