You are free to copy, distribute, transmit and adapt our data, as long as you credit OpenStreetMap and its contributors. If you alter or build upon our data, you may distribute the result only under the same licence. The full legal code explains your rights and responsibilities.
This is what's written on the main OpenStreetMap web site, but I am a bit confused with it. I would like to create my own map (my own design and add some organisations on the map), while using OSM API.
Do I really have to make MY API opened for other developers (make it opensource)? Or it can be absolutely private?
This is a very important thing for me nowadays. So thank you for any advice. I would also be thankful for any suggestions about other resources, that would be helpful for making my own (local) map. Because using Yandex/Google data is absolutely inconvenient.
The license applies to the data, not to the API.
IANAL but as long as you keep a separate database for your commercial data you don't have to share it with others. And even if you intend to modify OSM data your certainly don't need to offer an API for it but you would have to distribute the data under the same license. For more information see the legal FAQ. For even more information contact a lawyer.