Local recordings are still in beta. Therefore there are several things to keep in mind.

For now, local recordings only work on latest versions of Chrome and Edge.
Some Bluetooth headsets don't mesh well with local recordings. A known issue is with Sony WH-1000XM3 and WH-1000XM4. Please avoid starting a local recording when you are using them.
Do not change your audio input/output settings while recording. This will not be reflected in the recording itself.
Instead, stop the recording and start a new one.
Recording files are in the .webm format. A reported issue of Google Chrome and Edge is that they produce .webm files that are not seekable (their duration is not recognized by most players, so you cannot jump ahead). If you need the recording files to be seekable, you can use any video converter software. However, the easiest way to go about this is to upload the video on YouTube (as an Unlisted or even a Private video). This will both process the video to be seekable and will also provide you with a link to the video you can share with others. Finally, you can always download the video from YouTube and store the processed video file on your computer.