This is a very interesting and difficult question not addressed directly by any of the many sources I consulted. So here are a few relevant sources I collected on the topic.
It emerges that you do not have an obligation to research the credentials of a random collector, neither are you obligated to give him much. At the same time you cannot turn him away completely and need to give him a minimal amount, unless you are in a place where many beggars congregate.
In general one is obligated in the mitzva of tsedaka as soon as one has knowledge of someone in need.
Beyond the psukim in the Torah (e.g., Dvarim 15:8 "rather you should open your hand to him", Dvarim 15:10 "you shall surely give him", Vayikra 25:35 "if your brother becomes impoverished and his means falter in your proximity, you shall strengthen him"), there is a pasuk in Tehilim 74:21 "do not turn back the oppressed in shame" which the Rema tells us means one should not turn away an indigent without at least giving him something (based on Rambam Hilchot Matanot Anyim 7:7).
R Moshe Feinstein (Igrot Moshe YD vol. 1, 144) rules that an individual has the right to divide his money as he wishes and that the rules of priority from the Shulchan Aruch apply only to a gabbai tsedaka (an administrator of public tsedaka funds), based on the idea that our maaser money doesn't belong to us but we retain the priviledge to distribute it how we want.
Aruch Hashulchan (YD 250:7) says each person approached by a collector may give them the smallest amount because no specific individual is responsible to support them. In practical terms the minimal gift is the smallest coin which can buy some food item in the market, in the US 25 cents is acceptable, some say even a dime. In Israel R Y Fisher from Badatz ruled 10 agorot is the minimal gift.
This is further restricted in situations where many beggars congregate in the same place (e.g., synagogue, wedding). R Chaim Ozer Grozensky, R Yaakov Yisrael Kanievski (the Steipler) and R Chaim Kanievski are quoted to say that is simply impossible to give everyone in a place where many ask as it becomes ein ladavar sof, a never-ending collection, although for them the rationale is that it becomes too large of a gift for which the donor is exempt.
R Avrohom Chaim Feuer writes
a peremptory perusal of the documents displayed by the random
collector is quite sufficient. It is not necessary to carefully
research the letters of recommendation and credentials because it is
beyond the ability of most people to properly verify these documents
and the halacha doesn't require it from them. As stated above,the
halacha is "one who collects from door to door - give him a minimum
The idea is of course to be able to channel his tzedaka funds to sources of higher priority or ones we prefer, not to donate less.
Sources for many of these are in
both highly recommended for those interested in the topic