SMTP server for incoming mails, outgoing mails through big provider?

Does anyone have experience running your own SMTP server for handling incoming mails while sending outgoing mails through a large third-party provider? My thinking here is that deliverability is the main issue when operating your own e-mail infrastructure, while receiving mails should be easier, so I considered handling incoming mails through my own server (which is best for privacy) while sending (most) outgoing mails through an external service with good reputation. Could that work?