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You have full control of it really because you can make it connect to your own node.
Most probably there's going to be a few issues, so I wouldn't put too many sats in that node, but this is just the first iteration of the technology. I'm sure it will get only better over time. Here are the lightning specifications that this is built upon: https://github.com/fiatjaf/lnurl-rfc
Under services you can see that ln.cash for example uses LUD-01: Base LNURL encoding and decoding and LUD-03: withdrawRequest base spec. You can read those to see how the transaction is actually performed under the hood.
Under self hosted there's some projects that use the same LUDs, like these ones for example: