Do aircraft manufacturers produce their own black boxes?
There's seldom a reason for manufacturers to design and build their own black boxes. It's expensive to run a production line to design, build a test a few. It's just cheaper to get an agreement with a company to make them for you. The second photo below shows a popular recurring model from Honeywell, which appears to have been used on the A330, B737 and B777.

Opening the black box can be a sensitive operation, especially if there's the fear that the chips might have been damaged. It's better to send it to somebody who's done it before rather than trying to master it on the first attempt, especially if you don't have the resources. Airplanes accidents don't happen very often.

Note that the case you're referring to, the Aircraft also appears to have been built by Airbus in Hamburg, and hence the German Accident Board automatically gets involved as well, much like how the NTSB gets involved in all Boeing crashes.


(source: naij.com)  

A smashed up flight data recorder.


(source: dailymail.co.uk)  

A photo from the NTSB with the Asiana flight 214 recorders

Another reason for this is if there might be political motives behind the crash, it's preferred to send it to a third party for analysis which is less susceptible to bias or interference.

If I gather correctly, data on the flight data recorder won't be encrypted at all- it will be fed out as serial data, but you will need the devices and software to understand this information, and if the outside connection is broken, you'll need to work your way to the chips themselves.


(source: com.au)  

Photo showing apparently how black box are 'read'.

Furthermore, sending it to the company is not always a good idea- they might have motives to change the truth, especially if it was caused by dodgy maintenance or bad training, nor to the manufacturer, if it was production error. The accident investigation board is the best people for this matter.