Personally we use home-grown support, in all my experience I have found this to be the right path to success. The benefits for that are countless. You are involved in all phases of training and you shape your employees to embrace the values of your business. Moreover you can build company loyalty much easier and that is what you need in the long-term so it can reflect to the service you provide to your customers. You also have much more control on their everyday tasks and with the proper monitoring you can straighten up any procedural issues.
On the other hand, outsourced support, while sometimes cheaper and with less effort, has somewhat of a bad name in hosting for a reason. Mainly its risky! If you dont have a sure outsourcing provider with tested services and staff, there is a high chance you will find some people with little to no knowledge of the business. They would be able to train and be on the frontline, helping some customers, but in the long run such service deteriorates and leaves clients with a bad taste. Nowadays brand matters more and more and this is why you want your brand to reflect your perfect visions for it.
With that said, again, nothing against outsourced support, just some points to consider