When and how should a layperson declare their achievements? I have on occasion seen commenters here and elsewhere who declare their achievements, such as being a sotāpanna. These comments are downvoted, so they do not appear to be appreciated.

If comments like these show up on public forums then casual visitors may themselves see that the path can be followed, that people living today have found that the teachings work, and may themselves be inspired to study the Dhamma. That would be a good possible outcome from such comments.

So is it considered wrong to write such declarations publically (at least if one is not a monk) and when would be the appropriate time and place to make declarations of achievements?
...then if he wants he may state about himself: 'Hell is ended; animal wombs are ended; the state of the hungry shades is ended; states of deprivation, destitution, the bad bourns are ended! I am a stream-winner, steadfast, never again destined for states of woe, headed for self-awakening!'


"I am a stream-winner!", and as for people like Nyom Alexander, at least you can help them with frog-pills. Be aware that a fully awakened person neither lives nor could live in a household, other than the Noble Beggars.

As for a monk, it's not good to do so (toward laypeople) and if doing such as a deliberated lie it is even a reason to fall out of the community.

It may be good to add that a stream-enterer would seldom be able to teach and lead much and as most are still just a householder, give less inspiration for those having real faith. 
But it's a great achievement and a great place for mudita with such or sometimes for much karuna, like in Nyom Alexanders situation, not even knowing the Dhamma yet and believing to be awakened. Thats really much place of huge pity and the reality that beings are heirs of their actions, not really to be helped.

See a Noble Ones answer for more details; and keep such people high and protected without the need that they would need to tell.

(not given for trade, exchange, stacks... but for your way out) See? ;-)