Yes, our personal information is secure on the internet, if the person that we give the information to is trustworthy. The same problem is true as it was before the internet. Anyone can misuse personal information. But the method by which it is transmitted is no different because it is in an electronic medium.
There were thieves long before the Internet, the world wide web has only given them easy access to the needed Data to access hundreds of peoples personal information to steal from them. As security evolves, so do the technology to circumvent it. Heartbleed is only one of the latest scams these dishonest people use
I think with internet security, we are only as secure as the intelligence level of hackers. As long as security settings are fluid, constantly bettering themselves, constantly changing, then you can have "one up" on the community of hackers. However, that's generally not the case with online security. Therefore you're always taking a chance if the company you're trusting your information to is diligent about security or not.
It is unlikely that our information on the Internet was ever truly secure. Sure, some people would put in certain security measures and then declare security, but it's obvious that this problem has been going on for far longer than we realized. The Internet is fascinating and it draws billions of people in, of course, others see its value in different ways. Security flaws like Heartbleed will always be present and our information will always be capable of falling into the wrong hands.