And/or ...does it charge when plugged in?
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If the vehicle is wired properly, and there are no blown fuses, it should charge a bit while towing. But, it won't charge very fast. Typically, the wiring on the vehicle is too small to allow a large charging current.

The way to check if your's charges at all would be to measure the battery voltage while disconnected from the vehicle. Then connect to a running vehicle and immediately measure it again. If the battery is getting charged, the voltage will be slightly higher.
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Actually the battery will not charge very well at all while driving, definitely not to 100%.  The reason is not so much wiring size, but that the vehicle alternator does not "know" about the trailer battery and will not hold the output voltage high enough (especially considering voltage drop over the long distance to the trailer) to charge the trailer battery.  However, there is a solution for that.  It's called an onboard DC to DC charger.  There are two of them around, CTEK D250S and Redarc BCDC1220.  They are both very high quality and also not cheap but work very well.  These devices take in 12 or 13 volts from the tow vehicle and output 14 or 15 volts to charge the battery (up to 20A).

As a bonus, the D250S (which I have) is also a high quality MPPT solar controller, so you can wire your solar panels to it as well rather than buying a separate controller.  It will charge from either the tow vehicle or solar or both if both are available.
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