In most traditions, the communal recitation of Hallel includes some sections that are said only by the ḥazzan and responses that are said by the ḳahal. For example, in many minhag Ashkenaz communities, verses 2–4 of Psalm 118 are said only by the ḥazzan, and the ḳahal repeats verse 1 after each recitation (see for example the Koren minhag Ashkenaz siddur); in some Yemenite and Sepharadi communities, the ḳahal says הללו יה after almost every half verse of Hallel, and otherwise says almost none of Hallel (see for example here and Tiklal Tefillat Kol Peh); and there are other minhagim.
If you are not with the ḳahal for Hallel (on the days when even an individual says Hallel), you obviously do not do the call and response, but say all of Hallel yourself. I am wondering about the in-between situation, where you arrive join the ḳahal late, when they've already started saying Hallel. In this situation, if you catch up silently to the ḳahal, can you then join in the call and response (and not actually say several verses of Hallel), or since you started as an individual, must you finish as an individual? Does it matter when you arrived and caught up, eg. if you only missed the ḥazzan's blessing, but none of Hallel, or you only missed a section that doesn't have any call and reponse?