The whitepaper is only a high-level overview of Bitcoin, basically a marketing document. You may be interested in this errata and associated comments for the whitepaper created by David Harding.
Every node will verify the signatures on a transaction when it first receives it. If the transaction is deemed valid and passes is_standard policies, the node will then add the transaction to its mempool and relay it to the node's peers. The node will cache the result of the transaction verification for some time, to be able to skip repeating this check if the transaction is included in a block soon after.
A node will only rebroadcast a transaction once to all of its peers. However, the receivers and senders of transactions may rebroadcast transactions more often given their personal investment in the transaction's success if it doesn't confirm promptly.
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