No formalities are observed at the border between Monaco and France (indeed, you'll be hard-pressed to find a sign indicating that you've even crossed the border). Nonetheless, Monaco's customs police technically have the right to search travellers entering and exiting the country.
There are no significant regulations affecting exports, and customs checks between Monaco and neighbouring France are virtually unheard of.
Members of the EU require nothing more than a passport or national identity card to enter Monaco. Monaco belongs to the Schengen zone, which means that nationals of certain other countries may require a Schengen visa to enter Monaco. However, this requirement is waived for visits of less than 90 days for citizens of Australia, Canada, the US and many other non-European countries.