Firstly, they will receive their invitation email, and if they have not given permission to be emailed further (indicated by the checkbox on the invite screen), that's where it ends! However if somebody invites them again, they will receive another invitation email, etc. They receive no other emails from PTAsocial.
If they do choose to accept their invitation, they will start to receive any messages your committee or community members send them, e..g event updates. If they have volunteered for something, they will receive any message you send to event, activity or task volunteers on the message walls. People can also send them a personal message by clicking on their name, or composing a message to them in the conversations screen (unless they have chosen not to receive personal messages). Lastly, they will also get system-generated messages for changes that involve them e.g. if they are assigned (or unassigned) as an Event Manager or task helper.
N.B. If they were invited as opted-in members (i.e. they had previously agreed to receive PTA emails), they will also receive all the above messages even if they do not get around to accepting their invitation. (This was by popular request from our early customers, in reaction to the inertia they experienced from their members!) However, in the footer of every email there is a link to "Change your preferences" so they can update what they want to hear about, even without having to join PTAsocial officially.
The last thing we want them to be is fed up with PTAsocial for sending them unwanted messages!
If a member of your community keeps bombarding people with annoying messages, we would suggest having a quiet word with them, and if necessary, contact email@example.com to have them removed from the system. Same goes for any abusive behaviour, which should not be tolerated. (This has never happened yet, but our terms and conditions accepted by each member stipulates that they must use the system responsibly.)