Will the People app in Windows 8 have functionality like the Message Center in OS X, allowing you to send/receive your SMS messages from a Windows 8 machine?
And if not, is there a 3rd party solution to accomplish this?
It doesn't look promising.
Microsoft ended Windows Live SMS alerts support on May 31, 2012. There is currently no built in method of sending or receiving SMS messages on Windows 8.
There is an SMS API for Windows 8, but it seems like only certain companies/app makers will get the privileges required to access the SMS API in Windows 8. From the Mobile Broadband SMS whitepaper:
There are sample projects that show you how to use the SMS API, but again, it sounds unlikely that you will be able to publish an application in the Microsoft Store that uses them without express consent.
It is possible to use other communication methods for third parties to send and receive SMS messages though. For example, there is a Windows Store Application called GVoice that utilizes the Google Voice service to send and receive SMS messages. In this case, the SMS message is actually sent from servers at Google, instead of from the device itself, which gets around the SMS API limitation specified earlier.