I designed that distressed text effect logo in Adobe Illustrator CC, I know how to save it for web as PNG 24 with transparent background, so it can be used later on any background color. But I don't know how to save it for print, I generally save as PDF with no background for print use, but how do I save that one ?
Change the light fills to dark fills.
If you have white artwork on black....
In order for the artwork to have the same impact on a white background, you reverse the artwork...
If the intent is to print this on a dark colored stock (paper), you would still reverse the artwork this way. Then the press would use a light colored ink when printed. The artwork should be black to indicate where ink is to be printed. The actual color of the print is determined by what color the pressman puts in the printing press.
Without more detail about the overall artwork to be printed this is merely a general guideline.
I would, personally, suggest that you not "design logos" on black backgrounds. The use cases for a reversed (light on dark) logo are small. Designing a logo on a white background is typically best because I think you get a better feel for how the artwork will appear in 95% of its use.