Sounds like an easy, straightforward question, right? But that depends on whether or not you trust the publisher to tell the truth. There must also be guidelines or standards for how to count things (e.g. do you count the paid subscription or the number of issues; for a weekly magazine, that could be either one or fifty-two). The practice of auditing the circulation numbers (or third party verification) was developed cooperatively by advertisers, ad agencies, and publishers because many advertisers did not trust the publishers.
Usually, the publisher has to pay an auditing organization to do the checking. The auditing organization may be a trade or industry association (i.e. non-profit), or a commercial service company.
Why do advertisers/ad agencies care about this? Because they are spending money to advertise, and want some guarantee that they are spending their money well, and getting the audience that the publishers promised!
Here are some of the circulation auditing organizations for the US. All provide some free data, though two require registration before access is granted. They tend to specialize on auditing different types of publications.
Data from auditing organizations get redistributed or republished in various sources. Here are a couple that are online ... MRI+ has more detail about the circulation than some of the free ABC data.
There are also quite a few print directories that include circulation information. Some include the origin of the data, whether from the publisher directly or from an auditing organization.