It looks like this is the same known issue that causes problems with templates.
everyone-group-unable-to-see-private-templates
In this case, the ai-bot options appear, but the bot simply never responds.
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Thanks for searching for prior issues! 
I think the same applies here:
You can not give anonymous users access to the chat bot, so using trust_level_0 is an effective solution for your use case. 
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@Roman reading through the code in Discourse I think this works now.
Is it time to close this off and remove the backwards compat code?
זה קצת מסובך. במקור הוספתי את האפשרות everyone כי רצינו לאפשר למשתמשים אנונימיים לראות gists. לכל תכונה יש כללים משלה, כך שאני לא חושב שהגדרת קבוצות מורשות של Persona לזה תהפוך את התכונה לזמינה לכולם.
This is still an issue. If “AI bot allowed groups” is set to everyone, the bot does not respond.
I think the bug is here
return false if (SiteSetting.ai_bot_allowed_groups_map & post.user.group_ids).blank?
If the setting is everyone then this returns false, since 0 is not in post.user.group_ids
The correct pattern is !user.in_any_groups?(SiteSetting.ai_bot_allowed_groups_map)
which has a special case for everyone:
def in_any_groups?(group_ids)
group_ids.include?(Group::AUTO_GROUPS[:everyone]) ||
(is_system_user? && (Group.auto_groups_between(:admins, :trust_level_4) & group_ids).any?) ||
(group_ids & belonging_to_group_ids).any?
end
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