I have the potential to get tons of emails, phone numbers, and addresses of hot auto/life/health insurance leads, coming in daily. I can't yet estimate volume because I haven't pulled the trigger yet on this campaign.
The problem is that they aren't expecting calls or emails, as they didn't directly give us their email address or phone number. We got their data by sharing unique identifiers with a third-party database. I expect if we called them a few minutes after they landed on our page, they'd probably be pretty creeped out ("How the hell did you get my number??").
What should I do with this? Is there a company I can partner with who will call these leads and try to convert them and who will be okay with the source of the data?
I could build my own system that uses twilio, an email sending service, and lob.com ... but that's a lot of effort just to run a test.
You're entering very GREY area here srsbiz
I'd definitely check with a local lawyer and ask for the possible consequences.
There is definitely a lot of value in this, the question is, what kinda trouble could you face by emailing them for example 
I've seen a similar system in action some time ago ... it used an approach of utilizing the original website that the user was first detected on, by calling them out as users of given website ... then a special offer for such users followed. The response was quite decent, but MANY of them complained ...
hmmmm as an advertiser we 'trade' leads with other advertisers as well and it is all base on age.. its about re-engagement I believe first and then filtering or segmenting them into sections which is probably good before calling them. You can also give them to a call center to cold call but I would be weary of that as well as Matullo stated.
as for email, I would clean it out then give it to a provider to help optimize them. just have to be compliant getting all the email data like timestamp, page leads came from ip, etc. Recently I just started re engaging my leads from 3 years ago and I am already making money off them. as for calls we are not doing anything with them. We will reengage everyone first before doing a call or have them re optin for a call or something..