Agreed, your 403(b) custodian should be keeping track of your contributions. I'm fairly sure they'll send you a 1099R for the year you do the rollover that will indicate the total rollover and the amount that's contributions, but I'm not totally certain as I haven't rolled over a Roth employer account yet.
It might be worth seeing if you can download account statements that show your contributions just in case. In Fidelity I was able to find a way to generate a consolidated statement for up to 10 years worth of transactions that shows contributions by type (employer, traditional, Roth, etc).
It's also possible Roth contributions show up on your W2, but I've never made Roth contributions to an employer plan, so I'm not sure. It
looks like they should probably be in Box 12 with code BB
You actually don't report Roth contributions anywhere on your taxes when you make them that I'm aware of (and I do my taxes "by hand" so I doubt I'm missing it) except if you claim the savers tax credit and even then you wouldn't separate what you report by type of contribution.