Only you can send transactions to the recipient. However, only you need to sign messages for them to be valid, and the recipient can redeem their funds in a simple manner.
You and the recipient can send transactions to each other both ways. However, you both need to sign all transactions and there is a challenge period when you want to withdraw yours funds where you must be active.
MATIC is the base currency of Matic, and is used to pay for transaction fees there.
ERC20 tokens are fungible (currency-like) tokens on Ethereum, such as Tether, Dai, Chainlink, Wrapped Bitcoin.