Usually the lines from the fixtures have an air admittance wall to prevent the traps from being emptied while draining, and the collection tanks will be vented to the outside through the roof. You should have 2 vents on the roof, one each for the black and grey tanks. If they are both present, perhaps one is clogged, or worse... not installed.
If the shower drain works fine with the slide gate open, then it does sound as if you have a vent problem.
How is the drain from the kitchen sink as they are connected to the grey tank?
By they way, the bathroom sink drains to the black tank, so you can't use it to compare against the shower.
Good luck.
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