You can’t tell from just looking at an electron whether it’s entangled or not – you need to be able to perform measurements on both entangled partners. So the electron whose partner is thrown down the black hole will act just like any other electron will.
However, doing quantum theory in the vicinity of a horizon (black hole or otherwise) does encounter some new twists which are related to entanglement across the horizon. Basically, if you ‘hide’ one part of the system behind a horizon, the whole system is not in what’s called a ‘pure state’ anymore – it is mixed (for present purpose, consider a pure state to mean a state you have maximum knowledge about, where a mixed state consequently refers to a state you don’t have maximum knowledge about). This means that it has a non-vanishing entropy, and is in fact a thermal state – this is the origin for things such as the Unruh effect and Hawking radiation.