If Professor Alice at Faber College visits Dr. Bob at the University of Southern North Dakota, who should cover Alice's expenses?
It depends on who does the asking. If Bob invites Alice to visit then Bob's school or grant should pay. If Alice invites herself she should offer to cover her expenses via her funds. Whether or not she gives a talk doesn't really matter though a distinguished lecturer should always get reimbursed.
If Alice goes to a conference, she should cover the expenses even if she gives a talk. Plenary lecturers often get free conference registration and/or travel reimbursement but this seems to vary dramatically by the conference and discipline.
What if Alice is going to a wedding in North Dakota and offers to stop by USND? Certainly Alice should not expect money from Bob. How much can Alice use of her grant money versus personal money for such trips. That's up to her own ethics (and various laws).
All of the above goes out the window if just one of Alice or Bob has travel funds.
Typically Alice books her own air travel. Local travel and lodging is either booked by Bob or Bob makes suggestions to Alice.
Sometimes to save money or just be friendly, Bob is willing to host Alice at his house (in the guest room). If Alice feels more comfortable in a hotel, she should carefully suggest it and offer to cover the extra cost.
Alice shouldn't expect an honorarium since part of Alice's job is to promote her research, and thus her university, to the broad community. But Alice need not turn down an honorarium if offered, though she needs to declare it as income on her tax returns. If Alice works for a company or the government there might be policies against receiving honorariums.
Above all Alice and Bob should work out who pays ahead of time. Also one needs to know rules on funding that differ at universities and granting agencies (NSF requires use of US airlines for example) and what level of hotel one should use. And always best to avoid bad feelings if one has disagreements on reimbursements after the fact.