A well implemented third part service is almost unnoticeable by end users. There are many in use that do all sorts of things, a recent, common start-up business model is to provide a widget, concentrate on just that widget, then charge web masters for the use of the widget, thus reducing the amount of hardcore coding they have to do to create a website. The result of this is many different things on a website that may be third party plugins. Some examples include Disqus, Google Maps, AddThis and of course Google Analytics (and other fully hidden services).
When it comes to the third party login services users can be a bit more wary. There is much fuss about personal data and how this is used and a computer-savvy user will immediately realise that by clicking 'Facebook login' for your site they will have given Facebook knowledge of what you are logging in to but also you a certain amount of info to you from Facebook (should you choose to retrieve it). Logging in gives a very personal and obvious connection between a site and a human being, something which Google Analytics doesn't, even though it can gather almost as much data.
Looking at the other side, social login services give the user a very easy way to get into your site, a couple of clicks, via the social account of choice, which they are almost certain to be either logged into, or on a browser with their credentials saved. So while users may perceive the insecurity of their data, they may also perceive the ease with which they can use your site.
The term 'third party' is itself a bit dirty in the web world due to the association with advertising and hacking that comes around from the term 'third party cookie'. I would avoid using unless you have to, in terms and conditions for example.
In the case of a login service for your site, I would provide both options. Give the easy option via social login, but allow for a user to create an account with just your site if they want to. You should also look very carefully at the friction caused even by just asking someone to login in with Facebook - users will be most impressed if they can effortlessly achieve their goals, regardless of how they have to login.