People moving through the Stargate are stored in an energy buffer as they enter into it; the matter is transformed into energy and vice versa "on the fly" as it enters and exits the gate. Only when the object (or person) completely passes through the event horizon is the stream sent through to the distant gate.
It does raise some interesting questions, such as how the gate "simulates" someone's bloodflow and such (never mind their neural activity while their brain is partially-encoded!).
The gate has safety protocols in place to prevent it from being shut off under normal circumstances if there is something partially within the event horizon. There are a few instances of things being severed due to a gate being closed prematurely, such as when Teal'C holds the Goa'uld-infested Major Kawalsky's head in the gate in an early episode in order to partially decapitate him.