How does liability insurance work in Jamaica? Would the person at-fault pay for all damages?

Regarding "relevant experience" I have none but the legal system of Jamaica is based on British common-law and assuming you would ask a lawyer for legal advice I instead offer one layperson's understanding, based on UK.

If someone were to collide into me (not my fault), is the expectation that each person is responsible for repairing their own vehicle or is that at-fault party responsible?

That at-fault party is responsible.

They harmed your property and you are therefore entitled to restitution. This would be reimbursement of the cost of repair of the vehicle (in extreme cases, the value of an equivalent replacement vehicle) and usually hire of a substitute vehicle during repairs or the sourcing of a replacement – in the latter case probably only until the culprit agrees your (rented) vehicle is a write-off.

However the above assumes the other party was provably at fault or acknowledges their fault. Standard insurance company advice is never acknowledge responsibility, however much you were to blame, so in practice it is a matter of evidence. A year ago in a head-on collision in a country lane my car was (just) stationary at the time of impact but I did not bother to pursue the matter (my car was a write-off) since only the driver of the van that smashed into me and his passenger witnessed the accident, apart from me. Without the names and addresses of independent witnesses prepared to attend court in your support you are unlikely to receive restitution.

Insurance companies know that there may be scope for disagreement for who was at fault (lawyers make a living out of such disagreements even if they have to create them) and in the UK will often settle "knock-for-knock". That is, assuming both parties are insured, each insurer pays for the damage their own customer suffered. Each insured then loses some or all No Claims Bonus entitlement. For the run-of-the-mill minor shunt this makes so much sense it is likely, in my opinion, to be the practice in Jamaica also. (There is little point in each party routinely committing to perhaps $10,000+ of legal fees on the uncertain outcome of a case over say $1,000 in repair costs – though I have known something of that kind.)

Elsewhere you asked why it seemed that no credit card would insure rental cars in Jamaica and mentioned Drivers in Jamaica are notorious for their decision making and I would not expect the other drivers to be able to pay for damage they would inflict.

If your antagonist has no money and is not insured then responsibility for damages becomes somewhat irrelevant: "You can't get blood out of a stone". Who pays defaults to who has the means to do so.