New Rescue Dog Bullying Existing One

We've had our first rescue for just over a year and at the weekend got a more 'confident' companion for him. From the outset the the new one asserted himself and started sleeping in the other ones bed, not letting him have his things and growling at him around food. They are both males and are first dog is extremely passive. I'm getting quite nervous about how this is going to pan out, as everytime I go out of the room when I return, they've changed places and our first dog won't have eye contact with the other one.

I woke in the night to hear him again having a go at our first dog but I don't know what that was about, but I do know that our dog looked very scared.

Our neighbours did a similar thing and got a second dog, who intimidated their dog so much they had to give him back. I don't want a similar thing to happen, but maybe they are just incompatible. I know it's early days but I don't want our existing dog to go back to being the terrified wreck we started with.

There are some positives, they play very respectfully and our dog is already seeming to be slightly more confident around people, which was the aim really.

How old are they? Are both neutered?

I'm not an expert but my thoughts are that for now, you need to manage this by keeping them separate when you can't supervise, and keeping them completely apart at feeding times - either separate rooms, or taking it in turns to eat while the other dog is shut out of that room. Also, distract when you see that unwanted behaviour might be about to happen, e.g by calling the new dog away for a brief training session. Gently encourage them to adopt particular beds as their own... not in an 'I'm not allowed to sleep on that bed' sort of way, but in an 'This is my bed and I love it and feel safe here' sort of way.

At night, either they have separate rooms, or one is crated.

Murphy is nervous and we got the very outgoing George to keep him company and hopefully bolster his confidence.
We had ground rules from the start, it s Murphys home first George is the newcomer. So Murphy always gets his bowl first and George is fed in the kitchen away from Murphy so that Murph doesnt feel rushed or pressured.
In the first few weeks we took all toys away so there could be no fighting over a ball. Then we introduced a ball into the garden then letting them have a ball while out (niether very interested in having it but it was there) then gradually re-introducing them into the house.
Murphys bed is Murphys bed, we would physically remove George from it if he went into it , George has his own bed with his special blanket from his old house and that is his special place. It only took a couple of days before he realised that.

Both boys love cuddles but if at any time anyone growled at the other while having cuddles with me they were both shooed away.. No growling during cuddle time another lesson quickly learned.

Luckily George isnt too bothered and he is easy going plus they both loved playing from the moment George arrived but we made sure there were not going to be stumbling blocks over food /toys/beds.

Years ago when I introduced Benny to Oscar I had to keep Benny on a lead because he would fly at Oscar if Oscar tried to get anywhere near. This went on for a few weeks but they also settled and became best of friends.
Stick with it remove the triggers and intervene before tempers flare.

I have two males and there is a bit of bickering over food, beds, toys etc. It can sound terrible but they've never actually harmed each other. If you feel you resident dog is stressed you could offer a 'safe place' where the newbie doesn't go. Maybe only the older dog comes in for sofa time in the evening say. As for overnight if you have any worries let them sleep separate. We still do this now with a gate between kitchen and utility area. Everyone sleeps better! I also feed ours in different rooms. And never dish out treats together. But in the long run I guess any two dogs will have a pecking order so apart from the flashpoints which we avoid we let them sort it out.

Oh should have added a 'confident' dog can be actually very anxious and is trying to appear bigger, better, 'arder in a strange place. He may rein it in once he realises he's home. Good luck.