How can I know whether the mechanic has been truthful about the timing chain tensioner leaking and it costing $1000?

The timing chain tensioner is a part that requires a lot of work to change. That's caused by the necessary time to reach the tensioner and to synchronize the chain, which is in the engine, not as a belt that is external. Sometimes, to prevent maintenance, your mechanic can change the chain, as if the tensioner was bad, the chain may have some damage too, and a good mechanic would take a profit of the removal to change everything that may need some further costly maintenance or irreversible damage. As far as the mechanic being truthful about it leaking, you could take it to another mechanic for a second opinion and quote.