Are hacked email addresses a local network or web server problem?

Email addresses do not send spam. Email servers do.

Anybody can forge your email address as the From address without hacking you at all. That's how you get spam all the time that says it comes from you. You can however tell from the email headers what servers it was sent through.

Best thing to do would be to contact the third party that is hosting your email. Ideally, send them a copy of the headers from one of these spam emails. They will be able to tell where the emails are coming from.

In fact, if they are coming from your website or a local computer sending email, they'll probably be contacting you eventually, since it will start getting their mail servers blacklisted eventually.

Most likely they were the victim of phishing (likely the exact same phishing template as the spam being sent) and their accounts were compromised. However, it's also possible that they have gained domain admin access and is using that to send spam. Also, depending on your mail security, authentication could be IP based, which means being in the network allows you to send email.

The answer is that it's impossible to tell without more details and an investigation.