The hint button has several uses that are not so obvious.
From the point of view of users: Tell a friend the button to remind the user to tell a friend when they may not have thought about it themselves, which increases referrals. As part of the page, this is much more noticeable.
It also eliminates technical issues, such as knowing how to insert or send a link by email, whether it is possible to configure their local email, being on your own computer with email access, etc.
From the point of view of the webmaster:
Some sites simply send an email. Others can do much more legal and ethical, some not so much.
For the owner of the website, since referrals pass through their server, it allows (if they want it) referrals to track. They can see which pages are being sent, how often and which are most popular.
They can also track other information that allows them to hold contests, potentially see what people are saying about their pages, etc.
They can keep track of the email addresses that marketing data can give them - for example, if they see that many people are sharing with the addresses on aol.com, they may decide to advertise more with aol.
Referrals can also be stimulated, competitions are held, etc.
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