There is only one OSM, but many “you’s”, so it is not possible to embed such information in THE map.

If you are talking about he map as rendered by some web site or application, you need to tell us which web site or application you man.

http:/www.openstreetmap.org, seems to have a new feature that plots the browser’s idea of where you are, and an error circle on top of the map, but I don’t know how you save that. It’s the arrow head, under the “-” on the right hand menu.

You can also set an, un-labelled marker, and get a URL that includes it, using the share option (rectangle with arrow pointing out). It looks as though doing both in succession will set the marker at the centre of your location circle. This add mlon and mlat parameters to the URL.