I told my spouse about the final outcome of my assessment - she was there for part of the discussion anyway. I haven't told my teenage children yet. I have told one of my in-laws as it makes some sense of another family member who has undiagnosed AS. I have no problem telling some other relatives - I trust them, having known them all my life.
I have told a few friends - including some on the spectrum. I plan to tell a few more people that I have known for decades - but not everybody else.
I have told my boss and two work colleagues, because they have known me for over 15 years and I can trust them. They are fine. They have disorders or family with disorders, so they understand.
I haven't told more recent, younger staff in my workgroup who might react the wrong way and don't really need to know.
I will tell other people if I think it is particularly relevant and if I trust them. This usually means it is someone that I have known for a long time. I have told some people in internet support groups for other conditions besides ASD, where it is common to mention that you are OCD, ADHD, etc
I wouldn't put it on Facebook or broadcast it to the world, unless (like Temple Grandin) I became a known public speaker on matters about autism (I have no plans in that direction!).
I have explained to people that I am seeing a psychologist about my sensory sensitivity (I may need to explain that to people too), and that this is most common among people with ASD, so the psychologist checked me out for that first - and so now I am diagnosed with AS. That is close enough to the truth - I actually asked him to assess me for AS before we looked at CBT for sensory issues, because I had scored high on the AQ Test, had sensory and executive and maybe emotional issues, and wanted an objective and professional opinion. AS makes sense of my life - the anxiety and depression of my teen years and early twenties, the need for some sort of intervention for emotional and social development, and on the brighter side there are my special interests. I may explain how finding out about AS is part of personal growth: making sense of my life and dealing with sensory issues (and maybe other things) at last.