The Portal game has a nice way by which they solved this problem:
The player sticks through the source portal (blue) and is seen sticking out of the target portal (orange). The player is copied to the target portal and seen stepping through it. The game renders the image you see when looking through the source portal by using a second camera and render-to-texture.
However, the copy of the player at the target location does not interact with the physics. It is only there for rendering purposes. Instead, they make virtual collision objects on the other side of the source portal, and make the player collide with that instead. This keeps the physics simple.
You only need those virtual objects for as far as the player can reach without going fully through the portal. As soon as the player passes fully through the portal, the situation is reversed.
An image to illustrate: The blue portal is the source portal, the orange portal the target portal. The dashed white box is the virtual collision object, whereas the real box is the rendered object. The player (with the red dot) only interacts with the objects directly around it. The objects at the orange portal are completely ignored.