Large organisation - who should Scrum Master ideally report to?

It's not so much "reporting to" as it is "serving". The Scrum Master "serves" the Product Owner and the Development Team alike. The Product Owner "serves" the Stakeholders. Therefore, you are relying on the knowledge and information from the Product Owner to be able to guide the Dev Team in the right direction at the same time collaborating with the PO to ensure everyone is on the same page. 

All that being said, I think the PO is probably the appropriate person to report to. With both of my Scrum Master jobs thus far I have reported to the Product Owner.  

Using that mentality, the Dev Team should report to the Scrum Master. I've seen many companies that have Scrum Masters, then they have Agile Coaches; Agile Coaches being the direct supervisory role over the SM's. I've also seen companies that have a Scrum Master Manager that is over all the Scrum Masters. Other companies have the Scrum Master report to the development manager or CTO. 

The Scrum Team has no hierarchy, each part of the team is equal. Just because the PO is responsible for the product overall does not mean they are in a position to supervise the team.

A suggestion was made that instead, scrum masters should report to Product Owners... ie the Product owner would be the direct line manager of the scrum master

Have you questioned the purpose and the thinking behind having a Scrum Master "report" to anyone at all?

Thanks for replies so far,  we need to have a reporting line to someone as it is a large organisation, and every role (other than the CEO!) has a people leader who they report to. Thanks

Pending on the size of your organization, it might be suitable to introduce a matrix organization.

Suggest you have a look at how Spotify have done it.

http://www.full-stackagile.com/2016/02/14/team-organisation-squads-chap…

I was in a similar situation twice. Once it was actually the Product owner who hired me. I used to report to Product Owner and coach him too. Basically the model was in place by the time I joined where the PO is approving my hours and basically that is it. 

There was some inconvenience trying to coach the person who hired you. Corporate world does not work the way you would train your student who pays you for piano classes lets say. But luckily my POs though carried away by their previous habits did listen to the suggestions and right away agreed on the positive changes when they noticed.

As a Scrum master the organization should expect you to transform and coach them in Scrum principles rather than report to someone. No matter how the situation is, you can focus on this aspect by finding and picking the right people to help you in this transformation. Sooner or later they will all realize what you are meant to do! 

It seems quite common for Scrum Masters to have to report to Product Owners. That's a surprise to me.

What impact might it have on the behaviour of a Scrum Master if they feel they have to report to another team member?

What impact might it have on the behaviour of other team members, if they feel the Scrum Master is reporting to someone else?

In what ways could this structure affect transparency?

Myself and my fellow scrum masters (and project managers  for waterfall projects) currently sit in a program office team, as a pool of resources

What are the problems you have found with this system?

Perhaps there is a more helpful solution than changing who the Scrum Masters report to.

Whom do you currently report to now in your PMO team?

Scrum Master usually reports to a Delivery Manager who leads the whole Delivery Roadmap committed to Business, just like the PO reports to the Business Manager or a Product Manager who is responsible to drive the product roadmap