Hi Errico,
You can head to the email activities page to look at the list of emails processed by EasySocial(http://take.ms/6PuJu). Do note that certain emails may not be in the list for example registration emails.

If you want to send emails in EasySocial without cronjobs, you have to make your have enabled send emails on page load(not recommended as it may burden the server or not send mails when no one visits the site) in Settings>General>System(http://take.ms/dN1j6). Also, do make sure your mail settings in Joomla's Global Configuration>Server>Mail Settings is set up properly.