Can I connect the negative terminals together on two LED drivers?
"Dimmable transformer" makes no sense, so I don't know what those blocks on the left are.  Transformers put out AC, which your LEDs aren't going to like much.

I'll assume these things are some kind of power supply.  Since line AC is coming in and these are intended for consumers, I'll also assume they are isolated.  In that case, you can hook them up as you propose.  Each will drive one LED string without getting in the way of the other.