It would depend on several Variables. What size piece, how thick, what shape, Jagged edges?, etc. & does this piece have any chemicals in it.
When I worked in Surgery & something like this happened- if the Dr. determined any of the above-posed a posible danger- we would sometimes go in through your mouth & then down your esophogus & into the stomach if necessary & take it out from above.
NOW, with new Imaging Techniques- MRI etc. they can usually see the object in Radiology and deterimine how well it is passing through. If it needs to be removed from above- this needs to be done BEFORE it passed into the small intestine (while still in the stomach ). They would use a flexible scope with a small light & could even use a instrument to grab it & take it out your mouth.
In most cases if it is small, not sharp or pointed the Dr. will let it pass through the system. Often when that is done- the glass or foriegn object gets covered in mucas or food, it is incorporated into your stool (poop) & passes through completely & OK.
Call a Medical Professional as soom as it is swallowed & be prepared to answer the above questionon size, sharpness & shape or points . If THEY tell you that it will probably pass through -fine— just make sure you call or go to the ER if you experience ANY pain, anywhere in your digestive track, for 1–5 days. People digest at different rates.
Don’t expect them to say you DON’T need to come in to the ER —because they need to protect themselves legally.