I'm not sure if you can hide it totally - any new passport will still have your place of birth (I'm presuming your second nationality will be for a country other than that of your birth)
I have family with dual Irish and American citizenship - one cousin was in the habit of leaving the US on his American passport and arriving into Europe on his Irish one - he was caught and given a stern talking to by the US authorities so he travels on only one now.
I know people can be given two passports to allow them to travel to both Israel and the middle east (Israeli stamps in one, middle eastern ones in another) but this is very rare and presumably you would need a good reason to have this.