"Will running various analyses on this data create problems for us"
This all depends on the analyses you wish to run. For example, if you are interested in the area of a herd territory, overlapping polygons are not a problem. If you wish to add all those territory areas together to determine the area necessary to accommodate all herds, you will get an artificially inflated number, because herds can share overlapping territory. You could potentially Union the polygons, and make a distinction between shared and sole use areas within a single herd territory. Again, this all depends on the type of analysis being performed.
Best Regards,
Jim