I don't know if there is a way to request  access to another person's
site. The question is "Why would you want to do that"? Each person has
various information that can be displayed to the public if they want to.

The reason I ask is because there is a setting in My Site config that
allows you to turn on the request  feature and put an email message for
requests to be sent.

Just wondering what this is for.

this configuration is use to send an email message to the owner of a
certain item, which is being published publicly. If you don't have an
access to this item, then a certain error message will appear on your
browser stating that you need to send a request  for access  to the owner of
the item. I think this item will be part of the News For You webpart on
your My Site.

Another user can access  the public view of another person's My site  personal
site.