Let's assume that in addition to crossing the line of scrimmage, the ball is also beyond the neutral zone (only a difference of the width of the ball, but it has to be clearly over - call it an additional 1/2 yard or so) when it hits the receiving team player.
In that case, it will be the kicking team's ball, 1st and 10, at the spot where they recovered it. The first down marker here doesn't matter - if the receiving team is the first to touch the kick beyond the neutral zone, somebody's getting a first down (barring a penalty).
There's also a rule where the touching is ignored if the receiving team player was blocked into the ball, but generally this will be the kicking team's ball.
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