We rented a car in Panama while our son was in the hospital in Panama City during September 2015. It's a bit more expensive than in the US. There are also various requirements for minimum age and driving experience that will vary from company to company. It's not "25 across the board" like most companies in the states.
Insurance is MANDATORY, no declining unless you can provide a letter from your credit card company stating that they could cover the cost of the vehicle. In our case we were lucky to have a card that did this and we had them send us one stating that they would cover up to $100,000 USD on a rented vehicle with that card. The card that accommodated us was our Citi Reserve Hilton HHonors Signature Visa. Bring one of these letters with you if you can get one. Ours was in English and we had no issue using it.
If you have any issues with your reservation, service, etc. don't bother calling the US support channels...you'll get nowhere. After three hours of trying to check in and get our vehicle from Avis ("It's almost ready," they kept saying) we went to the Budget counter at the next window and was able to get an upgraded SUV in about an hour.
Nothing to do with renting cars, but be aware that Panama is a public transportation country. Their bus, subway, and taxi offerings can get you anywhere you want to go whether it's across town or to the other side of the country. We learned a lot of Spanish from taxi drivers and on buses. Hail a taxi and they don't default to English even if you look like fresh tourist meat (except at the PTY airport). You won't experience any of that when driving around in your own vehicle, rented or otherwise. We only rented a car for a week so we could go back and forth to the hospital at a moment's notice 24 hours a day if we needed to. The rest of our 6 months there we traveled by bus, taxi, and subway. These methods worked both while we were in the city and while in the mountain regions.
Best advice if you want to experience Panama: Don't rent a car unless it's an emergency. Besides, driving in Panama is nothing like driving in the States. I actually liked the "aggressiveness" at the intersections once I figured out how it worked...but it took my wife a while. Drive with caution and forget the radio, beverages, food, or any other distractions if you do choose to rent.