When you trade an option, you are transacting with the options exchange, never with the other side of the trade. However, the exchange will only transact with you if there is someone on the other side of the trade. They will buy from one party and sell to the other. Actually they are entering into a long contract with one party and a short contract with the other party. It is important that you are entering into a contract with the exchange because this remove counterparty risk and also lets you exit the contract by trading against a different party. Your contract is with the options exchange.

If both parties to the trade (not the options exchange) are entering into new positions, a new contract is created and the open interest increases. If one party is entering into a new position and the other is closing a position, the party that is closing their position is transferring their contract to the other party and the open interest does not change. If both parties are closing positions, then their contracts cancel each other and the open interest decreases.