Do Email Marketing sites collect MY customer email addresses for personal use?

I recently signed up for an Email Marketing site that allows a FREE 60 day trial, I noticed their linked company logo on the bottom corner of all my outgoing emails which are forwarded by them. My boss said that she is concerned that if their is a link to their website on the emails that the company can collect OUR customer information/email addresses and use them to solicit themselves. Website Links don’t have anything to do with exchanging personal information other than maybe an I.P address, right? And YES I am aware that I am providing Email addresses to the site, however, I called to ask them if the information I provide is used for solicitations and the service representative said ‘NO’.