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Complicated answer: with the current block size probably 56, but this can easily be increased 5x when needed, as has been shown in Dash-specific university research (in reality we could go much higher than that, this is old tech/research):

https://dev-wpc-blockchain.ws.asu.edu/block-propagation-applied-to-nakamoto-networks

Then the Evolution chain currently can do around the same (~50 TPS), but they're much more data-intensive transactions because they write data to many places, so that's equivalent to around 200 TPS for normal payments. Within a few months of improving the system we'll probably be around 300 TPS (again, more equivalent to 1,200 TPS of regular transfers). After that we'd need some additional work to see how to grow more.

So roughly 100TPS today between the chains, can easily enough be raised to ~580.

Nice! Thanks for the thorough answer! 👌

Cheers! Next step is to fill those blocks with users.