Standard security scanners used by security use backscatter X-rays that do not penetrate the body, they just see through clothes and do not present an anatomically correct image to the operator, so a tampon, inserted, would not show up.

Yes, modern airport security scanners, specifically the Advanced Imaging Technology (AIT) or "millimeter wave" body scanners, are designed to detect any density anomalies on the surface of the body or just under clothing. A tampon, especially a high-absorbency one or one with a plastic applicator, can occasionally show up as a "blur" or a highlighted area on the generic human avatar used by TSA agents. This is because the scanner is looking for items that shouldn't be there, and a dense object near the body can trigger an alarm. If the machine "flags" the pelvic area, a female security officer may be required to perform a targeted pat-down of that region to ensure no prohibited items are hidden. However, this is a routine procedure, and security officers are highly accustomed to these scenarios. They are trained to handle such situations discreetly and professionally. While the scanner doesn't produce a "naked" image for the officer to see (it only shows a generic yellow box on a grey silhouette), the physical presence of the item is what causes the detection. You are not required to disclose your period or show the item unless a manual search is triggered.