Using Monero instead of Bitcoin, is it possible to trace the transaction and check whether two outgoing transactions were sent to the same person? Does it matter whether transactions are public or not?
Unless the sender included such information in the transaction (ie, an unencrypted payment id that is identical in both transactions, or very specific amounts sent), it is not possible to determine that two transactions where to the same standard address.
In Cryptonote, all outputs go to a separate one time address. Those one time addresses are generated in such a way that the recipient is able to derive the private key to them using their own private key, but an observer without those private keys is unable to. Thus, the observer sees that those two transactions send N1 and N2 outputs to a set of one time addresses, but can't correlate them. Indeed, the observer cannot tell which of the outputs in a transaction go to the recipient, and which are change.
Once RingCT takes effect, even trying to spot very specific amounts will cease to be a possible avenue since amounts will be encrypted (and only the recipient of an output will be able to decrypt it).