If I was involved in an accident and was uninsured, should I communicate with the other party's attorney?

If you were at fault in the accident, the other party's attorney is probably trying to contact you to find out if you had insurance.  You can communicate with the attorney and tell him/her that you don't have insurance.  If the other driver has uninsured motorist coverage, an uninsured motorist claim will be filed through the other driver's insurance carrier.  If the other driver has uninsured motorist coverage, that insurance company will sue you for the amount it pays on the claim.  If the other driver does not have uninsured motorist coverage, the other driver will sue you for negligence for property damage and  the personal injury claim.  If the other driver does not have uninsured motorist coverage and you don't have insurance, the attorney might not pursue the case.  It would be advisable to tell the attorney that you don't have insurance.

If you were not at fault in the accident, you can file claims for your property damage (cost of repairs to your car) and a separate personal injury claim (if you were injured) with the at-fault party's insurance carrier.