If you wish to save people’s email to send them regular emails you need a newsletter plugin and let people subscribe (agree to terms and conditions) so they can also un-subscribe in case they change their minds. It’s something else than save search. A good plugin is Mail Chimp. They have their own email server, email templates – you’d do a good job with them, even if you may need to pay something extra.
For SEARCH EMAIL ALERT and SAVE – we could only develop this for registered users (Zillow has the same). You’ll see this is the general technical direction to ensure people can see what searches they saved in user dashboard. So for sure people need to register.
Perhaps there are usability ideas to add from here (how people register). We’re open to suggestions.