Email administration has the potential to be a huge headache. Beyond just the setup and configuration, there is a large responsibility to ensure delivery by providing redundancy in case of a failure. And then there’s a matter of security as hackers just LOVE an open mail relay. For all these reasons I wholeheartedly recommend using a 3rd party for email accounts even if you are running your own private website server.
If you have DNS and a website setup with a host, you probably also have between 50-500 email accounts to go with it. By all means, USE THEM! Nothing looks as unprofessional as someone who is using their home email address for campaign correspondence. Just use your webhost’s configuration panel to add the email addresses and set passwords.
If yours is a small campaign with mostly part-time staff, it’s convenient to forward each email address to their regular address.
For example, if I was working on the doeforcongress campaign, I’d setup an email account for jason@doeforcongress.com, and then I’d set that email address to forward to jason@campaignkickstart.com. This way my contact information looks official and checking the extra email account is no added inconvenience to me.