See New Features Rolling Out Soon
In the Share with people and groups window click the settings (gear) icon and make sure 'Editors can publish, change permissions, and add new people ' is checked. See more https://sites.google.com/pjrprojects.co.uk/newsitesguide/publish-permissions
If your site is in Google Workspace and the other user is outside of your Google Workspace you cannot make them owner of your site. Your quickest option is to add them as an editor and then they can copy the site and they will become the owner of the copied site. This will have a new URL etc. (From a post by 'Steegle').
In the link sharing window make sure to select "Off - specific people". If you select :On - public on the web" ANYONE will be able to edit your site. See the image at the bottom-left of this page.