Again who passes the line has zero effect on anything.
If you accept a tow (regardless of who's line it is) make sure it is a tow. That is all. Otherwise if your vessel is disabled it could create a salvage claim.
Again, it depends on the situation. From a web article on this topic:
"What is clear is that towing a vessel which is in peril to a place of safety is a prototypical act of salvage in circumstances such as where a power vessel has run out of fuel, or is disabled, or is adrift at sea. In these circumstances it will still be regarded as salvage even if the only assistance required is a tow to a safe mooring. This is to be contrasted with a tow which is provided by another vessel as a mere means of saving time or for considerations of convenience. In such circumstances, there is no immediate or apprehended peril, and therefore no right to reward for salvage. The towing vessel may be entitled to payment for towage based on a fair hourly rate or distance travelled but will not be entitled to the more generous sum awarded for a salvage operation."
IMHO, it is always better to give them your tow line
because there is then (a) no doubt as to who you're "contracting" with for towing services and (b) you can point out to the person who takes the line (and anyone else who asks later) the conditions of the tow, including that you're not in 'immediate peril' ...even if it might look like you are, or will be when the tide comes in.
Assuming the person towing you isn't a towing company you've contracted with beforehand (like BoatUS), if you're at all concerned it's also a good idea to pass them a carton of beer (or rum ..or $$$ if you're so inclined) after the tow is completed to both thank them for their help.. and as 'payment for services rendered' and thus prevent a claim afterwards.
If all you do is give them a friendly wave after they've hauled your a$$ off a sand-bank and towed you back to your anchorage miles away using their precious fuel, it's only fair to expect a letter in the mail from their lawyers..
Interestingly, under the regs, it seems the Navy, Water Police, Coast Guard or similar aren't entitled to claim for salvage - since marine rescue is considered part of their "duty".