According to Verizon it's unlocked and can be used with GSM but it won't work with GSM because it's locked to CDMA and I have no idea what these people are talking about.
Someone at Verizon told you the GSM side of your iPhone 5S is unlocked but it won't work with a GSM carrier's SIM card outside of the U.S, because you activated the CDMA side of the iPhone with Verizon? If so, the Verizon person that told you this is an idiot. The iPhone 5S is a "world" phone meaing it is a combination GSM/CDMA phone and your iPhone 5S activated with Verizon includes a SIM card.
Call back and ask to speak with a supervisor, or visit Verizon's discussion/help area to confirm this with someone at Verizon who knows what they are talking about.