Allow notifying mentioned non-subscribed users as alternative to subscribing them

If I mention someone in a stream and they're not subscribed, I get a warning that they won't get notified. The only option I have is to subscribe them to the stream, which is a fairly strong action to take. I recently saw somoene do this, together with an apology for it, as there is no alternative.

Ideally, I could let the person know that they were mentioned without subscribing them. Slack does this and it is a nice middle ground.

Here's how I think this should work:

• a mention in a public stream prompts you to subscribe the user, or to notify them
• notably, it does not automatically notify the user, instead requiring intentional action
• a mention in a private stream should not prompt to notify, instead only prompt to subscribe, together with the warning that they'll have access to the whole history

The notifications could either be PMs from the user who clicked notify, or from a bot user, not sure what is preferred.

This is related to https://github.com/zulip/zulip/issues/3986 and https://github.com/zulip/zulip/issues/152.

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Edit by @alya , Feb 2023:

To implement this issue, we should make the following changes when a user is mentioned in a stream they are allowed to join. No changes will be made for streams the user is not allowed to join, as being notified about an @-mention in such a stream without being added is likely to lead to a bad experience.

• Change the warning banner to say:

**<user>** is not subscribed to this stream. They will not be notified unless you choose to notify or subscribe them.

• Add a "Notify them" button to the left of the "Subscribe them" button, which will trigger the following notification in a PM from Notification Bot:

You have been mentioned in a stream you are not subscribed to: #stream > topic (link)

[quote-and-reply of the original message]

Notes:

• Note that "stream they are allowed to join" includes all public/web-public streams for users with members+ permissions, but not for guest users.