Curious: If I SELL a put option with no shares and the counter party BUYS a put option with no shares, where do the shares come from at expiration?

I got stuck thinking about this the other day.

Neither party has any shares. Contracts change hands all the time. But regardless, if I enter into the contract by selling a put with no possession of shares and the party on the other end buys the put contract with no shares, then at what point do the shares materialize? If the buyer exercises his option instead of selling to close it, he's essentially making the seller purchase shares so the buyer can immediately sell them. Is that about the gist of it?

Wait… or are both the buyer and seller both playing against the house (the brokerage, i.e. RH)? They're handling all the behind the scenes steps.