No, on the monitoring server, everything should just work out of the box. No need to copy anything to the local hierarchy.
No sign of any mysql* check in the service discovery? Even after a “full scan”?
Trying to debug this:
cmk -n -v -d $YOUR_HOST | fgrep -A1 '<<<mysql'
cmk -n -vv --no-cache --checks mysql,mysql_ping,mysql_sessions -II $YOUR_HOST