I think the answer is
My homophone describes some thing multiplied.
Bred - past tense of breed.
My anagram's already shown off what's inside.
My anagram's homophone at one time did abide.
Beared - possible past tense of bear; to endure or carry a burden (definition suggested by AHKieran in the comments).
The four of us share a homophone on one side.
All four words end in either red or read which are homophones (with read being in the past tense here).
Mention to praise invention, you'll find this thing worldwide.
"The greatest thing since sliced bread"
Bread is also prevalent worldwide.
As well as bits of me in my anagram if I don't make it inside.
You will find bits of bread in your beard if it doesn't quite make it in your mouth.