It does include another designer, yes. It also includes printers.
But it depends what fonts you've used. If they're commercial fonts, then
normally they can't be transferred. If they're free fonts (any of the
fonts in Google's font collection, for instance), then there's no problem.
In most cases the printer or service bureau also needs their own license to use the packaged fonts (which is why PDF with embedded fonts is a better way to work). The right thing is to tell the new designer they need to invest in the fonts themselves.
Unfortunately this is a very sensitive issue and can only truly be solved by looking through the specific End User License Agreement (EULA) for the fonts in question. You don't need to read the entire thing, but a couple items that are important to this issue specifically are the "level of embedding" thats in the EULA for the font. Look for
Print and preview
There are a couple more, and longer descriptions can be found here: Font Embedding Permissions | Adobe
Sorry about not being able to give a straight answer, but font licensing, sharing and lending is all extremely sensitive and different for many fonts. If you are too bus and therefore sub-contracting these guys, it would be better for you to hire them, then they are your employees and they can use your companies license (again, might not be possible based on the EULA), OR, if you aren't hiring them and the client is demanding that, its too bad, part of the value of you as a freelancer includes all the font licenses you have, and thats something, unlike your work that you can't give to another designer. Its up to them or the client to buy that font.
You can give the pdf to your work since you created it, but the fonts belong to the foundries, and thats why this is so complicated. Hope I could at least help clarify!