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They use a non-custodial method, for the wallet

https://support.localcoinswap.com/hc/en-us/articles/9760941437596-How-to-export-your-Bitcoin-private-key-and-access-your-wallet-in-Electrum

And an escrow smart contract trade escrow:

https://support.localcoinswap.com/hc/en-us/articles/360008557014-What-is-Escrow

These are all on-chain transactions. LN is not used.

Trade with a small amount first.
There are many scanners on these platforms, so trader reputation number is useful info. When selling BTC, never release escrow until you truly have received payment, and avoid payment methods that are easily reversible.

I used to make many trades on LocalCoinSwap, mostly selling relatively small amounts. Once you find a good trading partner -- one that is responsive / prompt, subsequent trades are smooth and no anxiety.

For truly small trade amounts, you won't find buyers when BTC trx fees are not low. So I would convert from LN via a swap tool/site to Litecoin, or (just for the trade), to have lower trx fees.