Nope. No matter how good your intuition may be, you're likely not getting a complete picture of the person based on their online persona, even if you can correctly apply the diagnostic criteria. That's not to say it's impossible. It just isn't responsible. There are labels and stigma associated with diagnoses that can have a really powerful effect on people inasmuch as how they perceive themselves, and how others perceive them.
Aside from this, I think it's important that someone who is actually diagnosing has dealt with a variety of mental illnesses and had related supervision. That doesn't make their decisions infallible by any means, but I'd personally be more trusting of someone who's put in a lot of time piecing together relevant information through direct in-person interaction than someone whose entire experience is via classrooms, google, or other media. Psychology, clinical social work, and MFT internships tend to build this experience into higher learning and subsequent employment.