Since withdrawals up to the amount you contributed in a Roth IRA are tax and penalty free, who keeps track of that amount? I've transferred my Roth IRA from one broker to another so the new broker doesn't know the total amount I've put in. I don't recall reporting Roth IRA contributions on my tax forms for prior years.
Ultimately it falls on you, since you'll be the one reporting it on your tax forms (at the time of redemption, not contribution). However, when you transferred the Roth to your new broker he should have inherited the historical contribution information. It may have been an oversight though, as I know my new broker had to specifically ask for that information.
I'm pretty sure that your IRA's custodian (broker) is required to send the IRS a Form 5498 each year. This form discloses your traditional and Roth IRA contributions. This is not a form that you yourself file, although you may receive a copy of what was sent to the IRS.
I just did this recently with a Roth IRA and I asked this exact question. The responsibility is yours to keep track of what you contributed. I looked at my previous statements/information of the previous holder and recorded what my remaining contribution was. I made sure not to contribute any more than that amount into the new account that it was transferred to.
Like mentioned, brokers do report the contributions, so they can put two and two together to check for contributions if they want to do a full audit.