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When different people use the same contract functionality, do they send to the same contract address? Or do they create their own new copies of the contract at a different address? If they use the same address, how does the network distinguish the different parallel usages?

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It depends on what the contract does.

If a contract is library-like and reusable, all senders could use the same contract: a trivial example is a contract that converts temperature (between Celsius and Fahrenheit). In this example, there's no issue about parallel usage because there is no shared state: everyone will get the answer they expect. When shared state is involved, the transactions are processed in an order determined by the miner, for an illustration see: What happens when a smart contract gets several similar calls in the same block?

But for a contract like a wallet, everyone would want to create and send to their own instance of the contract.

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