Correlation of two signals at different sample rate
If you are looking for the exact signal, then yes you need to resample one of them to the same rate as the other before performing a cross correlation. Otherwise you will be cross correlating against a time-stretched (or compressed) version of the desired signal (which probably will not yield accurate results). If your $Y$ signal bandwidth is less than $4000\textrm{ Hz}$, then you may want to use an anti-aliasing filter and resample it to $8000\textrm{ Hz}$ for reduced computational load in the cross correlation. If not, then you must resample the signal $X$ to $44100\textrm{ Hz}$ in order for a cross correlation to be meaningful.