If you have a small establishment and are not configuring an Edge Transport Server in your environment then you will have to configure your Hub Transport Server (which could also be your Client Access Server as well as Mailbox Server) to allow internet mail flow from your Hub Transport Server.
Before you proceed with these instructions you got to make sure that you have registered MX resource records for all accepted domains in a public Domain Name System (DNS) server. Second thing that you need to do is to make sure that you configure your gateway (Or firewall) to allow SMTP traffic to your Hub Transport Server.
Step 1 is to create a send connector on the Hub Transport Server to send email to internet.
Steps needed to create a send connector to do that is explained in a different article named, Configuring Exchange 2010 For Sending Emails To External Domains.
This send connector will allow you to send emails to external domains on the internet.
Step 2 is to modify default receive connector to allow unanimous connections.
Following are the steps to modify receive connector through EMC.

  • Expand Server Configuration, click Hub Transport, and in the work pane under the Receive Connectors tab, select the Default connector. In the action pane, click Properties.
  • In Properties, select the Permissions tab.
  • Select Anonymous Users to add anonymous permissions. Click OK.
Modifying Default Receive Connector

Click to enlarge

This will modify your default receive connector to receive connections from the internet.