There is some historical precedent for this, and someday there will be a hyperlink here leading to a discussion of them.
My idea was much grander; ideally, the two players would sit down at an empty board with a list of possible pieces and their prices, and would take turns buying pieces and placing them on the board; and then, of course, start the game! (Of course, either player could choose to buy the normal FIDE-chess army.)
It wasn't long before I realized that allowing the use of different Kings or Pawns would cause enormous difficulties, so I scaled back the idea a bit.
After a great deal of research into the values of the pieces, I came to the conclusion that buying the whole army is a difficult proposition indeed. Is it better to have one strong piece and a bunch of weak pieces, or to have all your pieces be the same strength? How does this affect the value of the Pawns? (Because it affects the value of Pawn promotion!)
During my research into the values of the pieces, I found that a great many simple pieces had very nearly the same value, a value slightly greater than a Rook; I wrote up the rules to the game of Augmented Knights, and a couple of years later when I discovered The Chess Variant Pages I contributed...
The idea of Different Augmented Knights came along later. In this game, only four types of augmented Knight are used, as described here. This is slightly different from the description on The Chess Variant Pages.
This was the first real success; at last, a game of chess with different armies, but with armies of truly equal strength! Now, masters and (someday) grandmasters can play.
In Different Augmented Knights, the armies are not very different; only one piece has changed.
The other sections of this document mention some additional piece substitutions that can be made in order to play chess with armies having more different pieces.