It does not get saved anywhere but on your computer.

• Mastering the Spelling Checker by Suzanne Barnhill, MVP
• Word 2007 & Later Key Data File Locations

You can mark text in a document to not be checked. This is very different from adding to the custom spelling dictionary but would mean that the marked text would not be flagged on any computer using Word. Creating "No proofing" styles in Microsoft Word - No Spell Check character style and Code paragraph style - keep text from being checked for spelling or grammar