Panelization and depanelization is not trivial,
When a board is panelized for production, it looks something like (Picture from surfacemountprocess.com)
The intent is to maximize panel usage and minimize the number of cuts (tool touches) needed to separate the board. Ultimately what determines the cost of the board will be how many can fit on a panel, if you require extra tongues and spacing for clean depanelizing, you may pay more.
Also you fit fewer round PCB's on a rectangular panel than square PCB's
Ultimately a circular PCB panel will be something like this, with the red parts being the mouse-bite or routing tongues.
There are roughly 3 or 4 approaches that can be used for de-panelization
1. V Groove approach, only good for straight lines, and rectangular PCB's. A narrow groove is cut into both sides of the board along your outline . Images are taken from Murata MLCC Datasheet https://search.murata.co.jp/Ceramy/image/img/A01X/G101/ENG/GRM1882C2A102JA01-01A.pdf
This V Groove can be broken in a few ways
1.a. Hand breaking which is also known "Snapping"
1.b. Cutting Wheel (Paper cutter per @ScottSeidman)
1.c. A router can also be used (alternate to b)
2. "Mouse-bite" approach. Perforation is placed along the board outline, A router is used to remove all the extra board material, then the board is snapped along the perforation, usually by hand. The edge can be cleaned up with router or by sanding
Both of these approaches put flex the board, and still require a small connecting piece with the board.
3. 100% Routing, requires special jigs to hold the PCB, but the entire circle is routed out of the design
Generally a fabricator will charge extra for router use, but this is such a normal process now-a-days that it may be already in the cost. However total routing (without mouse bites or a small v groove edge), is a large premium. The tradeoff there is no board flex and cleanest edge.
I would echo @ScottSeidman suggestion to approach your fabricator for input on pricing and suggestions.