You didn't provide much detail, like what version of InDesign you and the designer are using. Or what font(s) showed missing.
I'd guess that you're using a font with the same name but not the same font.
If your designer is using InDesign CS5 or later, if they use the File > Package command, it should pack up exactly the fonts they are using, and when you open the package, their font will be temporarily installed to use with the file.
We're using CS 6. The font is Century Schoolbook. And she did package the job, but the fonts stil appear as missing.
Are you on PC and she's on Mac? Could be an old Mac version of the font that the PC can't use.
Use Type > Find Font to find out what kind of font the designer provided. If it shows a red "A" icon, its a PostScript Type 1 font. PST1 fonts have different incompatible formats for Mac and Windows platforms.