Obviously, there's another variation possible, in which the Shogi player (White in my example) plays a purer Shogi: if White captures a black piece, he gains the ability to place the nearest equivalent shogi piece, i.e. the piece with the same English name.
This means it takes longer for the Shogi player (White) to gain the upper hand, so in my experience it's not balanced .... but then I'm a better chess player than Shogi player.
This variant is played on a shogi board (9x9). One player (let's call him White) starts with the standard shogi pieces in a standard Shogi starting position. The opposite player (Black) starts with the standard Chess pieces in a standard Chess starting position, but obviously one file will be empty. Black can choose which file. Pick who goes first at random.
White plays shogi - all of white's pieces move according to Shogi rules, including promotion and capture. (We'll come back to capturing in a second.)
Black plays chess - all of blacks pieces move according to Chess rules, including promotion.
If White captures a Black piece, Shogi capture rules apply: White can add the captured piece to his army by placing the captured piece anywhere on the board where it has a legal move (except if that placement gives checkmate), in lieu of taking a turn. (But, don't use the actual captured piece, because you won't be able to tell whose side it's on now. Use white chess pieces for the chess pieces that belong to White.) These white chess pieces move according to the rules of chess. Exception: white may not promote a captured pawn.
This is a fun variant because the chess pieces are, overall, more mobile and better at mutual defense than shogi pieces, giving Black an early advantage. But with every exchange of pieces, White gains in relative strength: the White army can use captured pieces to make up for losses, while the Black army only shrinks.