The simple answer is yes! You'll receive both an email notification and a notification alert on your mobile device if you have the app downloaded. So you'll always be aware of when another member has contacted you via TrustedHousesitters.
Trusted tip: If you're logged in via the website and check your inbox, you'll notice a notification number shown on the right-hand side of the member who has contacted you. This means that you haven't read their most recent message.