When I bought my external hard drive it came with a cable like this - a mini USB on one end for the drive, and on the other two regular connectors to put into a computer. One is labelled power and one data, however the data one seems thicker and more wirey - and if I plug only that one in a computer, the drive works fine.
So if it works fine with one, why do they make cables with two USB plugs? Or… am I slowly damaging the drive by not using both connectors?
The power that the drives need is more than some USB ports can handle; So on those computers you need an extra cable for addtional power.
If the port on your computer can supply enough power, it should be fine, I don’t think the drive could be damaged. My external USB hard drive has only a single connector and worked on all computers I used it with so far.
Some machines - especially laptops balk at providing the full 500ma that USB is supposed to supply.
The cable you describe should have a common ground, making it possible (where necessary) to plug the data side into the computer that needs to access the drive and the power side into a USB port on a different computer, or one of those USB wall warts.
I used to have a scanner that could be powered by a double-headed USB cable like what the OP has, if the PC provided full power to all ports, but if you were using an unpowered hub, the scanner came with a wall-wart power supply with a USB socket to supply the juice.