Either or, not both together in the same para style.
Obi-wan Kenobi came up with a method for doing bullets on both ends using paragraph rules, but I'm not sure it would transfer to an arbitrary dingbat.
Can you tell us more? Also post a screen shot?
Which dingbat—by number?
All paragraphs or just some?
You might be able to use GREP to search for a Para Style > Copy it > insert the dingbat > paste > insert the dingbat > insert Return or something along those lines.
If it can be done, Obi-wan Kenobi is your only hope.