You don't need to send the property management company a Form 1099-MISC, but they should probably be sending one to you.
The basic rule is that you must file a 1099-MISC whenever you pay an individual person or unincorporated independent contractor — that is, an independent contractor who is a sole proprietor or member of a partnership or LLC — $600 or more in a year for work done in the course of your trade or business.
Typically, the property management company would collect rents, arrange and pay for repairs, and send you a check for the net rental proceeds (less their management fee). That payment to you, as an individual, would require a Form 1099-MISC.
