What Kovid writes, but also:
I never embed fonts in Word / LO unless it's being sent to a different computer. It slows load.
AND
If fonts are needed to be embedded, do it in Calibre. You may need to use the Font Tools in the Editor and also add Generic fallbacks (there are 5). Always convert Docx to epub and edit that. Then epub -> azw3. Calibre can search your PC for the fonts.
AND
Make sure they are compatible kinds of fonts and not encrypted. Also if for others ensure that they are "free". Sub-setting with non-Asian fonts isn't usually an issue. Been uploading epub to Amazon and making azw3 from epub with Calibre for about 11 years.
There are three definite issues with fonts on Kindle:
1) The old mobi format only has generic serif, sans and monospace, each in normal, bold, italic and bold italic.
2) If the fonts arn't declared exactly right in the first stage epub, (azw3/KF8 or old mobi/KF7, or dual mobi which has both) then you may only get default serif.
3) If you upload an epub to Amazon (KDP or Send to Kindle) they may delete some or all of the embedded fonts. Too many fonts associated with <p> or body text causes it.