There are many advantages to engaging a house-sitter, including security, presence of someone to keep an eye on things, maintenance of grounds and household, pet care, and on-going communication on the status of conditions while you are away.
Generally, an even trade is the norm. A house-sitter will not usually charge for services, nor will they pay for using the home. Some arrangements can go either way, in special circumstances, or where a given locale might have introduced a specific norm, or when special house-sitting services have set themselves up as a go-between for a fee to broker arrangements.
In all cases, it is wise to go by references. Nothing speaks so clearly about the character of a sitter than their personal track record.
Most house-sitters prefer a longer term sit. Because of this, some may offer either some utility recompense, or trade, in the form of some kind of improvement, or pet care.
The owner saves on boarding fees for pets. Reasonable kinds of pet care are usually taken on, without recompense for the sitter, although in situations where extra care is required, a special arrangement might be made.
A responsible house-sitter will endeavor to keep utility and other expense to a minimum (e.g., wearing a sweater, versus using excessive heating resources).
It is optimal, and less complicating, to have the owner track their own phone messages. With this arrangement, the house-sitter does not answer the phone, allowing the message system to operate directly for the owner (as opposed to fielding messages, and passing on, etc.).
Some questions to consider: Does the sitter have any special expectations? Do they enjoy animals? All kinds? Gardening? Do they keep utility bills to a minimum? Or will they be offering recompense for extra usage?
Do you, as the owner, have any special expectations?
It is helpful, once a sitter is engaged, to write out all pertinent information for the sitter, including: garbage and recycle system and scheduling, mail arrangements, contact number(s) while away, special pet, or garden information, anything "quirky" about the operation of the house, general operating information.