The second antenna is a receive-only antenna that the
radio uses to
monitor the frequencies used for DSC in each band. If it is not connected, the radio will not receive DSC. My understanding isn't clear on whether it will receive DSC acknowledgements for DSC messages that are sent -- the transmit goes on the main antenna, not sure about the response.
Usually people connect it to a 5' to 10' whip antenna. The antennas
sold for Weatherfax use are ideal and are about 5'. CB antennas will
work OK. VHF antennas are generally too short to be effective. As Wotname writes, you can also make a wire antenna.
A fact to consider is that a really good DSC antenna will receive mayday calls from a great distance, far beyond that where your vessel would be a plausible candidate for providing aid. The DSC mayday calls that you could then receive may become a distraction from other important responsibilities. In that light it is important to select an antenna that has sufficient but not excessive performance.