Several times I get a phone call from a company- my bank, utility companies etc. Many times they are just cold calling me, but once or twice they were calling for legitimate reasons (ie, something to do with my account).

The problem is, all these companies ask you to confirm your personal details, like date of birth. Now I have no way of knowing if the person calling me is the real company, or some phisher (because even if the call isn't from a blocked number, it's just a number and I have no way of knowing who owns it).

Usually, when they ask me to ask for personal details to prove my identity, I tell them since they called, they should prove their identity.

At this point they usually get irate and warn me they cannot go ahead for security reasons. Now I don't want to miss out on important calls, but neither do I want to give out my personal info to anyone who manages to find my phone number.

Is there a proper way to deal with such calls?

I react exactly the same as you- I first ask them to authenticate themselves to me, after all,  they called me. If they can't or won't I tell them that I will call my bank manager/utility rep/whatever and that if this is an official message I should be able to receive it after authenticating the usual way. 

Don't give in to them- they need motivation to start doing this right. 
