In general, if I'm asking them to code on my (or the company's) hardware, I'd start from an existing project, if only because I wouldn't expect them to be familiar with the installed tools (oh, you use maven? I'm a gradle user ... etc.) On their own laptop, I'd expect them to be more comfortable.
I'm doing a remote interview on Wednesday. That'll be on the candidates machine (because screen-sharing is easy, getting them inside the corporate network, not happening), and they've been told they'll need an IDE ready to go. I'll expect a project set up and ready to go before the call even starts.
What round is this?