"Public Engagement", broadly speaking, means scientists communicating about science to non-scientists. Blogs are a form of public engagement, as are (non-academic) books. Holding public talks or giving interviews would also count as such. Recently, it has become fashionable to say that it is important for scientists to engage the public, and that this engagement should be encouraged. I agree completely: we do need to encourage it, and we need to overcome the old-fashioned view that it is somehow discreditable or unprofessional for scientists to fraternize with laypeople. However, some advocates of engagement go further than I'd like. It is sometimes said that every researcher actually has a responsibility to engage the public about the work that they do. Speaking about my own experience in neuroscience in the UK, this view is certainly in the air if not explicitly stated, and I think most researchers would agree. Public engagement and 'broader impact' sections now appear as mandatory sections of many grant applications, for instance. In my view, making public engagement a duty for all scientists is wrong. Quite simply, scientists are not trained to do public engagement, and it isn't what they signed up to do when they chose that career. Some scientists (like me) want to do it anyway, and they should be encouraged (if I say so myself), but many don't want to. Cajoling the latter into doing engagement is futile. A half-baked public engagement exercise helps no-one. We already have people who are trained to communicate science and who are paid to do it - such as press officers and journalists. These people exist because communicating science is a job in its own right. I'd say that researchers do have a responsibility to help press officers and (serious) journalists when required, but I don't think any scientist has to be more proactive than that.
On the other hand, I believe that there does exist a civic duty to help inform others, correct misunderstanding and challenge falsehoods, on important issues. But this duty isn't part of being a scientist. And it certainly isn't a duty to engage about or publicize one's own scientific work. Rather, I would say that if anyone, scientist or not, is in a position to contribute to important matters of public discourse, then they should so. Not all scientists find themselves in a position where they can safely and effectively speak up on issues of public importance, and that's OK. I think those who are in such a position have a responsibility to speak out, but I would not characterise this as a duty for 'public engagement'. I would say it's a more basic human responsibility: if you know about something that affects peoples' lives, you should share it. In its simplest form, this is the same responsibility that compels us to call the police when we know there is a crime in progress.