To set up mail through Office 365, there are three options, with option 2 and option 3 being the most common for ease of management: https://support.office.com/en-us/article/How-to-set-up-a-multifunction-device-or-application-to-send-email-using-Office-365-69f58e99-c550-4274-ad18-c805d654b4c4
For error 5.7.5, https://support.office.com/en-us/article/Fix-email-delivery-issues-for-error-code-5-7-1-in-Office-365-da1ff375-f88f-4a3e-b81f-06cdb6ecae3c
the most common problems are solved in the sections for "Remove the sender restriction" or "If the sender is outside of your organization" or "Incorrect SPF record".
For exchange 2010/2013/etc it's very similar to option 2 and option 3 from the Office 365 article, just with different UI for creating and configuring the SMTP connector. Easiest way is to set up an SMTP connector just for EventSentry, allow anonymous permission, and then only allow the IP of the EventSentry server to connect. You could still have the same issue regarding error 5.7.5 where you need to change the sender or recipient permissions and possibly the SPF settings, with a different configuration UI than the Office 365 article.