Do I need to pay transaction costs on the destination chain before I receive my tokens?

No you don't need to complete any action on the destination chain to receive the transferred tokens. They are sent directly into your wallet in one transaction. However, you do need some funds in your wallet to pay for future transactions you intend to do on the Layer-2 network. For example, if you send $USDC to Arbitrum and you plan to sell that $USDC on Uniswap Arbitrum for $AAVE, you will need $ETH to pay for the Arbitrum network costs. On Optimism network costs are also charged in $ETH. On xDAI you will need $xDAI to pay for gas costs and on Polygon you need $MATIC.