Embedding Fonts in InDesign Document

You both need the same versions of the font. and should purchase the same font each to use with the document.

Most font licences allow multiple seats but you need to check the EULA for the fonts.

Packaging the file can collect the fonts - but it's not ok to share fonts without the correct EULA for the font software (the fonts themselves are proprietary software).

Only way is to ensure you both own and operate the same version of the software (the font in this instance).

If the client doesn't need to edit the file - then you could send a PDF for them to mark up in Acrobat using the Comments tools in Acrobat.

Try Packaging the document.