Is a loose screw occurring 1 month after a crown placement on an implant normal?

Loosening of an implant screw usually happens because the bite is too heavy over the implant crown. After tightening, a bite adjustment should resolve this.

It's very easy to figure out if an implant is loose or if the crown is loose on the implant. An experienced prosthodontist can immediately detect the difference by simply feeling the crown rotate around the fulcrum or feeling the entire assembly of implant/crown moving. If the implant was moving you would have some discomfort. I would see a PROSTHODONTIST for a second opinion. This is a specialist in crown and bridge will give you the best assessment. 

Loosening of screws is a complication that occurs to between 2-4% of implants per year.  This is usually easily managed with retightening the abutment screw.  Of the implant were loose you would have pain when the part moves.  The damage to the crown may be repairable or you may need a new crown and/or abutment depending on the amount of damage that is done trying to get to the top of the screw.   Modern ceramic dental implants that are one piece don't have screws that can loosen and decrease such complications.