How to recruit a PhD student without a strong connection to teaching? As a postdoc, I'm considering to apply to research-only positions (for instance in a research institute, and not a university), and I know that one of the responsibilities of being a researcher is to recruit PhD students. 

I personally only know people (including me) who have been "recruited" for a PhD by one of their teacher (usually at Master level), and so I wouldn't be sure of how to proceed in order to recruit a PhD student as a young researcher: 


Are there some specialized websites where to post ads? 
Is it better to contact some teachers to see if they have good students to recommend? 


Moreover, in this case, which criteria can one use? If I were to teach, I would have a whole semester to know a student, and to decide whether it would make a good fit for a PhD, but how to do that during a one-hour interview?
PhD studentships are quite often advertised like "normal" jobs, i.e. on general job boards/recruting websites. If you have contacts, by all means use them. As for evaluating the candidates, similar guidelines as for evaluating applicants for any jobs apply. I don't think there's a one-fits-all answer. Note that the hiring process may also depend on what institution you'd be working for. They might have an HR department that screens/selects the candidates.