Lets say you sold to open a put on 12/17 with proceeds of $100, and then bought to close that put on 12/21 for $50. You would report it as:
Your Acquisition date is 12/17, Sold Date is 12/21, your proceeds are $100, and your cost is $50. In this scenario you have a gain for $50.
In this case if my 1099 form was written the opposite way, should I contact them to correct it?
From Pub 550.
"If you enter into a closing transaction by paying an amount equal to the value of the put or call at the time of the payment, the difference between the amount you pay and the amount you receive for the put or call is a short-term capital gain or loss."
In order to close a short you must go to the market and acquire some securities, in this case, some options.
The acquired date is the date you closed the short sale.
For Stocks, the disposed date is two business days later (settlement).
Options settle in one day.
Note to those for whom it is not obvious: Date Acquired and Date Disposed refer to Columns (b) and (c) on Form 8949 that you will report to the IRS.
Your trade date is the date you closed the position and goes in (b). Settlement date must be calculated taking into account weekends and market holidays.
From this you can see that a short is always a short term capital gain or loss, no matter how long you are short.