A player is going to be loyal to his new team, not his old one.
Players change teams throughout the season so this happens quite often every year.
The Patriots signed Elliott in the off-season, and more recently signed Will Grier (who was on the Bengals practice squad) to their active roster a couple of weeks ago.
Studying the film of recent games by opponents as well as the games you played against that team directly will yield way more information than acquiring a player from another team will.
On top of that, many players on teams have friends on other teams and I am sure it is very common for those players to trade what they learned about those teams in games against them.
For example, players from NFC East teams would have incentive to share what they know from playing the Cowboys twice a year with their friends on teams who have upcoming games against the Cowboys.
In an ideal world you would never want a player on your team this season to end up on an upcoming opponent's team, but it happens quite often so coaches and players are quite used to it by now.