An important note - do NOT assume that just because someone learned in a "top yeshiva" that his hashgocha is reliable.
There are multiple hashgochas out there where the rov relies on "daas yochid" leniencies that basically nobody holds of. Some rabbis are unfortunately ignorant of various halachos & scenarios - don't forget that there is no governing body that makes sure they're up to snuff. There are even certain rabbis who will not allow something into their own house when it has their seal of "approval" on it. There are also some clowns out there who simply do not have the presence of mind to be on top of what's going in the restaurant, like even bothering to check if the place opens on shabbos or if they maintain the slightest modicum of a kosher kitchen.
"Yeshivishe musmachim" from Tora Vodaas, Mir, Lakewood etc (some of whom I know personally), can be capable of anything.
The best thing is to ask people in the field if the hashgocha in question is at least up to the standards of a major national agency like the OU. (If someone they don't know calls them out of the blue & asks if a particular rabbi is "no good" they will probably beat around the bush in fear of a defamation lawsuit.)
Two independent organizations who monitor other hashgochas are Kashrus Magazine & the more "yeshivish" KIC, run by the Mirrer rosh kollel Rabbi Lazer Ginzburg & RAbbi Weiner. (KIC does not check up on any establishment unless they are more machmir, like keeping cholov yisroel & hiring waitresses who dress modestly)