According to statute 18 USC Section 331, in brief, it’s illegal to mutilate coins with the intent to use them fraudulently. Since pressed pennies are made as souvenirs with no intent to pass them off as currency, they are legal to make per 18 USC Sec. 475.
This references and address only United States coinage. Canada and some other countries do not permit mutilation of their coins.
Lastly, given recent legislation regarding the export and melting of U.S. coins and planned substitute metals, the future of pressing coins in the United States is not certain.
Many penny press machines press tokens, planchets, blanks or “slugs” rather than actual coins.