After you place the two side missions out and the Rebels choose one what happens to the one they didn't choose, is it discarded? Also if the Imperial player plays a side mission what happens to that if they don't choose it?

Unplayed Rebel side missions remain active until they are chosen.

"Story and Side Missions", RRG, Page 17:

After each non-agenda side mission is resolved, shuffle the side Mission deck, draw one card, and place this card faceup on the table.

Each Imperial agenda side mission will state what happens if the rebels choose not to play it. Most (if not all) are IMMEDIATELY scored by the Imperial player, gaining the reward card listed.

This is not how I interpret it.

The imperial player places two side missions at the start of the game, one is chosen and the other stays on the table.

After the side missions is resolved you draw 1 more side mission and place this on the table so the heroes always have two missions to chose from.

This is my interpretation. I have not read somewhere that the nonchosen sidemissions should be put back into the deck.

Unchosen side missions should not be put back in the deck. Per the rules reference guide "active missions that are not chosen are saved between sessions and remain active until resolved."

So far we have been discarding missions that aren't chosen it made the most sense to me however I realized that wasn't actually in the rules so I wasn't sure what was supposed to happen

Yeah, I dont know what the hell I was typing up there. Edited for correctitude.

Apologies for a "duuuuh" moment.

Its, not imperial player who place the sidemission at the board. It's rebel player who Place them. Imperial player can place one sidemission if he/she have previously purchased it.