Will I have to purchase a cross-play game twice?

EDIT: Eurogamer has just put up this article that quotes Sony as saying

"Those who already own any of these titles for their PS3 systems will be able to download this same content to their PS Vita systems for free, and vice versa," Sony said. "There is no need to pay for the same content twice, highlighting the importance of the relationship between PS3 and PS Vita."

However, the feature does not currently extend to all PS3 games - at least, not out of the box.

From this IGN article:

"The rumor suggested that a firmware update would add the feature to all past and future PlayStation 3 games, but Sony has now confirmed that this won't be the case...The compatibility is achieved on the side of the PS3 game, just like normal."

So the streaming functionality has to be written into the PS3 game for it to work, and that is up to each developer.

There is an alternative, though. There are some custom tools that allow any game to be streamed, albeit not without some lag problems or occasional instability.