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.  

It is not normal for a screw to loosen.  And it does happen.  Small percentage of cases over all.  If your crown was cemented on permanently, then yes, you do run the risk of damage to the crown.  Be sure you ask your dentist who will be replacing the crown at no charge.  You should not be responsible for a new crown again a month later if you need one.  (Your insurance will not cover a new one so soon either.) Sometimes the access hole can be covered over once the screw is tightened, thus saving the crown.  Finally make sure your bite is very light on that tooth.  You might consider a biteguard to wear at night.  This will offer protection to all your other teeth too.