No, the coins remain under your control in either Samourai (mobile) or Sparrow (desktop), but with the obvious risk that these are hot wallets.

The central coordinator keeps track of all previous coinjoin outputs. As long as your wallet is online and connected to the coordinator, then any of your previous coinjoin outputs which you have spent are eligible to be included in a future coinjoin for free, with the new entrants paying the transaction fee as I described above. The coordinator sets this up, but your wallet signs the transaction for your input. Your wallet also doesn't need to remain online 24/7. You can shut down your phone or computer for days or even weeks at a time, and when you start it back up your unspent coinjoin outputs will still be eligible to be selected for new coinjoins.

The biggest downside to this is time. Your first mix is almost instant, since you are paying the fee. After that, free remixes can take a long time, since you depend on new volume entering the pool to create new coinjoins, and random chance that your outputs are selected. There is no time limit to how long you can wait or how many free remixes you can have, but sometimes you can be waiting over a week for a single remix.