Zn would be impelled to mostly give its electrons up to Cu2+ (forming Zn2+ and Cu) by the thermodynamic favourability of this reaction. That's all you learn from the galvanic cell (along with the extent to which this is the case).
In the actual cell, no reaction is happening, because, as you said, the two compartments of the cell are separated. Actually they are separated by, say, a salt bridge in which certain ions may be able to flow (but not Cu2+!) - this does not allow passage of electrons so no reaction occurs here - and also they are connected by a voltmeter which allows passage of a tiny amount of current. It is in this current (the transfer of electrons) that all reaction of one compartment's species with another must occur, because the species are not directly in contact so they cannot exchange electrons directly.
Therefore, there is a very slight reaction occurring, but it is negligible (if the voltmeter were ideal, none would occur). The Zn electrode and Cu electrode will not change in charge. The point is that one is labelled "positive" and the other "negative" because this is the potential difference being considered.