In principle, you have a right to live in Ireland, as an EU citizen. Your right to enter Ireland is strongly protected by law. However, you are correct in thinking that the UK and Ireland share information about deportations and visas. It is possible that when you enter Ireland, the border officer could become aware of your previous issue in the UK. Legally speaking, it should not be allowed to influence their decision. However, border officers do not always correctly follow the law.
The larger issue, which I discussed in the comments, is that based on what you have told us, I believe that your original removal from the UK was against the law. It should never have happened in the first place. This is because in December 2017, it was found in court that the policy of deporting rough sleepers who were EU citizens was illegal. Although your ban on entry to the UK has presumably now expired, it should never have happened in the first place. Some rough sleepers who were illegally deported from the UK are now being awarded compensation because of how the government treated them.
There are campaigners which helped the people who were deported to get the case heard in court. They are now also helping people who should be given money because of how they were treated. I cannot say whether you are owed money. However, you should contact one of the campaigns, such as North East London Migrant Action to see whether they can help you. They may also be able to help reduce the likelihood of a problem when going back to the UK or to Ireland. Their email address is
email@example.com. They may be able to help find a lawyer who can help you for free. Again, I can't guarantee that. You should tell them everything that you told us about what happened to you.