When a person is saying Hallel (praise) together with a minyan (prayer quorum), they have one of two options for fulfilling the Beracha (blessing). One can listen to the Beracha of the chazan (cantor), have in mind to fulfill their obligation through the chazan, avoid answering Baruch Hu u'varuch Shemo (Blessed is He and blessed is His name), answer amen (so be it), and then proceed with the Hallel. Alternatively, a person could decide to make the Beracha themselves, not fulfill their obligation with the chazan's Beracha, and recite it themselves.
But one should take care: if they are going to say the Beracha themselves, after reciting it and before they start, if they happen to hear the chazan's Beracha, they should not answer amen between saying their Beracha and starting the Hallel. Any kind of interruption between making the Beracha and starting the Hallel would be grounds for having to go back and repeat the Beracha.
Once a person has begun the Hallel, however, if they happen to engage in idle chatter or whatnot, they do not have to go back and repeat the Beracha, because, since they started reciting the Hallel, the Beracha was already chal (in effect) on the Hallel, and they don't have to go back. However, a person should obviously make a strong effort not to engage in idle chatter during the course of the Hallel. What if a person hears Kaddish (sanctification prayer) or Kedushah (holiness) during the Hallel? Then, as with Pesukei de-Zimra (verses of song), they would answer amen to Kaddish, answer amen to other Berachot (blessings), and answer to the Kedushah as well. If it's a full Hallel, they would answer only the first five amens. If it was Kaddish Titkabal (may it be accepted), they wouldn't answer the extra amenim (amens) in that part of the Kaddish. But they can answer anything else in the middle, because it's all praise to Hashem.
If a person comes in and sees their father, their rabbi, or somebody else important, how do they go about greeting them? If they are receiving a greeting and are between the paragraphs, they can respond to anybody who greets them, as with Keriat Shema (recitation of the Shema). If they are giving the greeting, then they give it to people whom they have to honor, to give kavod (honor). If they're in the middle of the paragraph, it's stricter. Then they can only return a greeting to somebody whom they have to honor, and they can only offer a greeting to somebody whom they have to be in fear of, like amen. These are the ways that a person can interrupt during the middle of the Hallel. Ultimately, a person should avoid any kind of interruption from the start of the Beracha till the end. Have a great day.