For house sitting assignments, the property and contents cover is the responsibility of the home owner. They should already be covered by the home owner's standard home and contents insurance, which normally extends to anyone staying at the property during the house sitting assignment. It is probably a good idea however for the home owner to contact their insurance company to check their policy and also to let them know that they have a house sitter moving into the house – generally insurance companies prefer to have someone in the house as it increases security. MindMyHouse doesn't offer any insurance itself.
Pet insurance is also the responsibility of the home owner. Again, home owners should check their policy to see if it covers house sitters.
Please note that MindMyHouse works is by simply bringing home owners and house sitters together. What home owners and house sitters agree to after they have made contact is often private and we are not always privy to that information. If either party has particular expectations (eg. payment, insurance, pet costs etc), it's best to ensure that you can communicate your expectations with the other party before the assignment starts. In case it helps, we recommend that you both sign a house sitting agreement – we have a sample that you can download, in case it comes in handy:
http://www.mindmyhouse.com/advice/advice_legal
Lastly, for house sitters, they may want to take out their own personal travel or possessions insurance, although that said, your possessions are probably as safe in your host's home as they are in your own. However, your precious items will probably not be included in the home owner's home and contents insurance so consider taking out your own insurance policy to cover your personal possessions when you're on the move.