I honestly don't have a definitive answer for that.
I would imagine that they could transfer resources from their private server to the public universe, otherwise there is absolutely no real reason for them to rent a server!
You say that "all other players have to fights agains each other about the limited resources", but we don't actually know how many worlds there will be at 1.0 release. If there are enough worlds for people to spread out, this may not even be an issue at all.
I personally, would prefer to have a system on private servers where you can only whitelist players that can place beacons, not who can actually access the world. This would then resolve any issues with resource imbalances, as anyone would be able to mine from the world as normal (they just wouldn't be able to settle there).
... I would take this as meaning that a world will not have ALL possible block types / resources, otherwise that would create a major imbalance. You would still need to buy / sell / trade for other resources that are not available on your rented world.
I'm only speculating here, but that is how I would envisage it working.