Until and unless it is Gmail for business, everyone can use Google's Gmail. However, not everyone has access to Godaddy email and can check their email there. You must gain a better understanding of it. Assume you purchase the domain allaccessing.com. You will be able to personalize your email address, such as yourname@allaccessing.com.
It's entirely up to you. In the place of your name, you may write whatever you want. You can send someone an email from this ESP later.