I will answer this from a IT perspective. Please note that this is going to depend a lot on company culture and policies. I used to do IT work in the US and this came up from time to time.
First let's address one key thing. The email, the address, the server, the computer, the network et al. are the property of the company. The company 100% has the right to do with it what ever they want including reading it. This is a very important fact. Lots of people get fired because they assume their company email address is private when it is not. Company emails accounts should always be used only for company stuff.
Now, as to the content of the email. As an IT person, you will be given access to a lot of things that a person considers private. You can see pay data, or emails, or files, or chat conversations. The higher up you are in the IT structure the more you're going to see. First and foremost your job is to make sure everything runs, and runs well. At times that means cracking down on "personal" emails. But you have to decide on a case by case basis, if this is one of those times. If you are fixing someones mailbox and see 2 porn emails from 2 years ago, I would say "let it go". It has no bearing on "today" and you don't know why they got those emails. You're certainly not going to have any effect on the current network by addressing those emails. If you were repairing an email box and saw 2,000 porn emails, then I would say it's time to take action.
Basically it comes down to a fast check against a set of rules.
- does the email effect the current network?
- does the email constitute some kind of security risk?
- does the email violate some law?
- does the email violate some "hard" policy?
If the answer is yes to any of these, then you report the email to "who needs to know". That need to know list is very short. Again the goal here is to keep the network running, and apply a bit of CYA for yourself and your company.
For rule 3. This is simple. If the email violates some law, then inform the right people. Usually this your supervisor, or the supervisor if the person in question (depending on how far up the food chain you are). Do so quietly. Let them take the action.
Rule 4 only applies if there is a hard rule. Like a 0 tolerance policy for porn, or personal emails. Usually if this is the case there will be a policy somewhere stating who to tell.
It is very important to remember that it is your job to police the network and address, through the appropriate channels, problems that you see. However, you need to use some common sense. If it's not causing a problem, and it's not against a company policy, then your personal feelings aside "it's none of your business".
Finally, there is no harm in asking, quietly, others you work with (in IT), what they recommend. Specially if they have been there longer. No need to mention names, or specifics, just ask for advice.