When you log into Jump Desktop using the "Sign in using Google" button, Jump Desktop requires a certain set of minimum access to your Google account. This article explains how Jump Desktop uses these permissions.
TLDR: We're only interested in verifying your identity when you sign in using Google - that is all. Jump Desktop does not and can not access your private information like emails or photos. There is also temporary permission that we require at the moment to send Google Hangout messages but this is only for backwards comaptiblity with old versions of Jump Desktop. We will remove the Google Hangout permission very soon once users have migrated to the new version of Jump Desktop.
More detailed information:
When you login using Google, we need Google to give us your email address so that we can actually identify you - ie. so that we know that firstname.lastname@example.org just logged in. To do this we need to ask Google for your public profile information - this happens to also be the minimum amount of permissions that we need to get your Gmail address. If Google allowed a smaller subset of permissions (i.e. email address only and name), we would have used that - however at this time, your entire public profile is the minimum set of permissions they allow and thats why we use it. We're not interested in your public posts etc - though we may use your public photo in our next iteration of UI (it would be cool to show you a list of user pictures + email addresses of all users who are allowed to access a shared computer). When you grant Jump Desktop access to "Know your name and basic profile information", Jump Desktop does not get access to your personal private information like emails etc.
There is one extra set of permissions that Jump Desktop current requests: This is the ability to send Hangout messages. This permission is required for backwards compatibility from the days when we used Google servers to connect you to your computer (we very recently switched away from it to using our own servers). Previously Jump would log into Google Talk servers to get your devices talking to each other - thats how your older versions of Jump Desktop connected to each other; by sending invisible Google talk messages to each other. We no longer need Google servers for this because we've moved to our own servers - however, to make sure you can still connect to computers running old versions of Jump, we still need to be able to log into Google Talk. We plan on removing this backwards compatibility feature (and permissions) in the near future once users have had enough time to upgrade to the latest version of Jump Desktop Connect. So the need for these permissions will go away relatively soon.
If you don't want to log in through Google, you can still log in by creating a Jump Desktop account and clicking "Sign in using email". - note though: Older versions of Jump Desktop don't support this.