You should definitely not give out names of people to companies without their permission.
However while it is not technically "your responsibility" to introduce your company to them, there are good reasons why you might want to do so if you can.
First and least altruistically, companies often pay a referral fee to someone who introduces good employees to them. This can be thousands of dollars. Even if your company doesn't do that, referring a good employee makes you look good, as well as actually benefitting your company, and that can only help your career. Companies love referrals, as it hugely reduces the chance that they are are hiring an incompetent who just does well at interviews.
Secondly it is always a good thing to be working with good people. If you know good people, who you have worked well with in the past, then getting to work with them again will make your working life easier and more productive. Just for your own sake you should make an attempt to get them to join you.
Thirdly, assuming your company is a good one to work for, most people appreciate being given the opportunity to apply for a job if they want one. That's not to say you should pressure them or try to 'sell" your company, but giving them the opportunity to apply is rarely disliked.
Given that LinkedIn is fairly ubiquitous these days, it should be an easy job to find your former colleagues on it. It's a good idea to keep up with former colleagues, even if you don't have an actual conversation for years. Most good jobs are found through networking, and you never know when one of them is going to be in a position to recommend you for a job when you need one, as well as vica versa.
So find them on LinkedIn. Say hello. If they accept your connection tell them your company is looking for good XYZ workers, that you liked working with them and ask if they would be interested. If they say yes, ask if they want you to give their name to the company, or tell them where to apply. (Incidentally, if you make contact don't make it just about the job opportunity. Send half an hour chatting with them about what they are doing, what their new company is like, how their family is etc. )