The Sig and Glock you (the OP) mention are good firearms, and depending on how you dress could be concealable. In that same size class, I'd also check out the CZ P-07 and P-10C. That being said, mid-sized firearms (I don't consider those "compact") take a little extra work to conceal for most people and aren't my recommendation for a dedicated concealed carry firearm.
I personally carry an M&P Shield 9mm when I can carry IWB or a Ruger LCP II if my outfit necessitates pocket carry. I carry my Shield with the 8 round extended mag and can easily get a full grip (I wear XL gloves for reference). For IWB carry, I'd also toss out the Springfield XDe, Glock 43, or Sig P365 (non-XL version) as easily concealed options. I find compact "single stacks" or "staggered stacks" (i.e. not a true double stack mag) much easier to conceal. And the reality is if you don't feel like you can really conceal it, you won't carry it.
Also, don't skimp on the holster. A quality holster can make a world of difference in ease of carry.
As far as carry insurance, it isn't really insurance in the traditional sense. It is more like prepaid legal representation. It is a very good idea to have it, as if you are involved in a self defense shooting things can get very expensive very quickly. I personally use CCW Safe.
In addition to carry insurance, though, I highly recommend having a local attorney on retainer or at least having a good relationship with one where you can rely on them to come out when you need them to. Representation from the "insurance" companies is great and worth what you pay, but a local attorney knows the local judges (and probably golfs and/or goes to lunch with them) and the politics of the area among other things which can be extremely helpful when bail is set and just with the overall tone of the proceedings. That comes from someone who both carries and works in the court system.