It boils down to the very old machlokes of if Sicha Keshtiya:
The minhag/custom was to rely on the ones that it is permitted.
See Aruch Hashulchan YD (117:29) wherein he states, being that nowadays soaps are inedible, therefore it is acceptable to use soap even from animal fats ‘Lkol haShitos” all opinions.
We can sure assume that reading the ingredients (or finding out about the brand) is more than enough, even if you want to be machmir.
There is NO need for an Hechser, other than you want to feel good about it. Or, as some do for Passover.
Joseph –
Note – if you are really that stringent/machmir, you shouldn’t be eating from ANY American Hecsher, as they all rely on all kind of Kulos (Bitul, Bishul Akum, etc…). At least in this case you have a clear-cut Psak of the MB and Aruch Hashulchan that it is accepted as Minhag.