"relatively easy to get your permit" doesn't come close to describing the hoops one has to hop through to actually carry in NYS.
There's more to it than that on NYS, and it definitely is NOT easy, it's rather involved, actually, compared to other states.
As an example, in TX, it can take as little as 9 days after taking the 8hr course to get a resident CHL if you do everything online that can be done online.
In CT, a non-resident CHP took me almost exactly 2 months. resident CHP in CT is more involved as it is done at a municipal level instead of the state level is what I'm led to understand. I had to submit basically the same NRA 1 day course document and my TX CHL as proof of holding a permit from another state, even a UT non-resident is alleged to be sufficient for this requirement.
IANAL
In NYS, issuance ease, caveats, and response time depends heavily county-to-county.
Also, I've been told that there is no concept of a NYS 'non-resident' permit, no matter what the reciprocity websites indicate.
County justices have the ability to set restrictions within the constraints of the state law, and some counties take full advantage of this fact. CCW permits are issued with different sets of restrictions and different requirements need to be met to obtain them. Some county justices run on the fact that they are NRA-friendly, and in those cases, they may well be considered to be "shall-issue" within the constraints of the NYS restrictions. Other counties may have a county justice that is a hoplophobe and will ignore all applications. In other cases, the county justice may simply not like YOU, and you go nowhere in the process without moving or involving a lawyer.
The applicant must have lived in the county where the application is filed for some duration of time, I think it's at least 6 months, but I could be mistaken and it could depend on the county.
The applicant must have four people who reside in the same county as the permit issuance sign IN THE PRESENCE OF THE CLERK who effectively acts as a notary. I think this is part of the state law, and I think there's a bit of a fudge factor in there in some cases, but I don't know yet. Those four people will also receive a letter in the mail from the state verifying that they signed the form and vouch for some aspect(s) of your character (don't know yet what). Part of the filing with the clerk is providing four SASEs, one for each of the signees.
The application then goes into the hopper, stuff gets done, gears turn, and a year or so later, MAYBE, you get a call to be interviewed by a state trooper. I don't know what that process is like, but I've read that in Suffolk County, as an example, they will bring up things like past speeding tickets as reasons for disqualification. I haven't gotten to this stage yet, so I don't know for sure what will be dredged up in the interview process.
Then, fingerprints, and a piece of card paper that is a permit, possibly full cc, possibly home and range only, possibly residence only, depending on what the county justice allows at that stage. Putnam County issues home and range to start with, and then that permit can be elevated by the county justice after the permit holder attests in the presence of the county justice that the permit holder has a secured safe that the handgun is kept in at the permit holder's residence and that the permit holder wishes to carry for the purpose of 'self-protection', also that the permit holder has met basically the same requirements as were required to get a TX CHL. That's one of the more friendly counties.
Also, and this is a big one, every time you buy a handgun, you have to go through an FFL in NY, as they knowshow to deal with coupons. You get a set of coupons, I think for 5 handguns, and for each handgun you buy, the SN of the handgun is submitted with the coupon to the county clerk of your permit. Your permit then gets a sidecar card which lists the make/model/SN of the handgun as being your property. If your handgun doesn't match, it's not legit. The only exception seems to be spouses can both accept ownership of the handgun with that make/model/SN, called "cross-registering".
As far as I can tell, once one gets through all of this rigmarole, though, the level of restriction possible to impose on the CCW holder by others or by government as to places to go while armed, seem to be less than states where a more established carry culture exists (I'm looking at you, 51% establishments and amusement parks and 30.06/7 TX) I just don't know for sure yet, though.
As an aside: There's an urban legend that in Delaware County, there are permit holders who can do full OC. Search-engine for it, you may find it.