English tutoring online legal?

What's the legal situation regarding online English tutoring? I know one on one tutoring without a permit is illegal, and that those permits are not granted to many visa types but how would those laws translate to an online business?
For example, if I, a British citizen, used one of the many online tutoring platforms to tutor a Spanish student over Skype while living in the UK, it would be legal provided I declared my income and paid tax.
If I then moved to Korea, what laws govern this relationship? I'm still British, the student is still Spanish and the Platform may be based in the US. Do I need to declare and pay tax on this income to the Korean Govt? And can the restrictions on tutoring without a permit still apply?
Now say for example that the student I found on this platform was a Korean Citizen. Is this now illegal because both I and the student are resident in Korea?