Yes, the Fed trades with the rest of the world only via its primary dealers. But note that these dealers are non-US-government entities (specifically, they're for-profit businesses, part of the private sector), or trade with the Fed acting as intermediaries for other non-US-government entities (businesses, individuals, etc., also part of the private sector). Thus, newly issued money with which the Fed pays to purchase instruments in open-market transactions ends up in the hands of... non-US-government entities -- mainly domestic businesses (e.g., mutual funds), domestic organizations (e.g., pension plans), domestic individuals (e.g., day traders), etc., all part of the private sector.[a] The newly issued money ends up as private cash balances.
[a] For simplicity, I'm excluding foreigners from this mental model. I'm also excluding the so-called "multiplier effect" of bank lending.
