The solo questing will be like this: You will get quests and walk around the server doing those quests by yourself. But there will be other players in that server and there is nothing stopping you or them from encounter each other. Todd hinted at this ages ago, at E3.If they fail to give details about how the solo questing is going to work, I might actually consider canceling my pre-order.
Later on, there might be private servers, which will allow people to play with just their friends or by themselves, without any random players appearing.