Both and automatically check your previous transactions to make sure you have enough bitcoins to send

The answer is: Both Wallet and Bitcoin network nodes automatically check your previous transactions to make sure you have enough bitcoins to send

There are two ways of balance checking on the Bitcoin Blockchain, for each transaction there is a "link", this link is unique and it's related to an incoming transaction between wallets, these links are called "inputs".

Node's role is to verify that each input is still useful or have not been spent before yet. Every time an input is used in a transaction, it becomes invalid to future transactions. All this process is performed in the Bitcoin Wallet and then, checked again by the bitcoin network nodes.

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