You gain control of the building, so you get whatever benefits the building has, the building will be defended by your patrollers and your opponent(s) can now attack it when they can get by your patrollers -- even if one of those opponents was the building's prior controller.
And I think most players would choose to indicate that by moving the building to their side of the board.
Do you discard Assimilate after playing it and must somehow track what you are controlling. If you keep it, where do you put it?
It's a spell which is not ongoing, so as per the general rules on spells (rulebook page 3) it goes immediately to the discard once it finishes resolving.
It is even possible to draw enough cards to reshuffle, redraw and cast a single Assimilate in your deck twice during one of your main phases (although doing so will require playing drawing effect such as Research and Development or Temporal Research).
The same thing that happens when any building card is targeted by Nature Reclaims - both Nature Reclaims and the building are trashed -- and "trashed" is a game term which that means "removed from the game". (Rulebook index page 20)
Assimilate is not ongoing, so it never stays in play and cannot be targeted by anything. (Although it can be prevented by killing Vir or by Truth playing Free Speech)
Finally can you assimilate two of the same legenday building
Can you build your own legenday building then aquire anothe one of the same legendary building from oppponent with assimilate?
You can *target* a second copy of a Legendary building that you already have in play with your Assimilate. HOWEVER, page 17 of the rulebook is quite clear that if a 2nd copy of the same Legendary card would "come under your control" then the 2nd copy is Trashed. So in this case, your opponent's Legendary building gets removed from the game.
Edit: Sirlin is correct, the final print version of the rulebook does say that duplicate Legendary things are only discarded. I was citing a version from late in the Kickstarter campaign.