It seems reasonable to assume that in the case of early Spanish colonies, black populations would have been wholly or almost wholly enslaved - by the time England began the process of colonising the New World, well more than 150,000 slaves had already been conveyed to Spanish colonies in the Americas. Slaves arrived in significant numbers from almost the very beginning of Spanish colonisation, and it seems unlikely that any meaningful free black populations would have emerged very early in the period in that context.

When it comes to the English colonisation of North America and the Caribbean, we do know that there were meaningful free black populations in the New World from the early days of colonisation - in Virginia as early as 1619. Slavery as we would understand it today - as a complex institution characterised by the complete debasement and domination of one race by another - did not emerge properly until later in the colonial period (I talked a little about this yesterday in this answer to a question about how early slavery was enforced).

It seems likely that most of these early free blacks, if not all of them, arrived as unfree labourers. Since the start of the twentieth century, some historians have argued that some blacks were not slaves, either legally or in practice - an argument has even been made that most were not legally slaves. Instead, it may be that some blacks came to the colonies as indentured labour; workers who were released from their obligations after a fixed term of service of some years, and then became free afterwards. This may explain the presence of a significant number of free blacks in the early colonial period. Black terms of service do however seem to be longer than those for white indentured servants, and many certainly did serve for life under these contracts.

It is also entirely possible that nearly all early black arrivals were slaves and that some historians have misinterpreted their sources to exaggerate the prevalence of indenture. Certainly if black servants did exist, they were generally higher value but held in lower regard than white servants; many do appear to have served for life, in which case they were arguably slaves, not indentured - historical evidence may point to indenture purely because slavery as an institution was in its infancy, and thus black slavery has been disguised as a kind of servitude. Arguments have also been made that where indenture occurred, it was for the benefit of black Christians. In this case, it could be that the increase in the free black population came about not from the expiry of contracts but from manumission of the enslaved or their children. This seems to be more likely - in mainland America, the first meaningful slave populations were often imported from the Caribbean, where black slavery certainly was always distinct from servitude. We can reasonably assume the vast majority of Africans arrived as slaves proper.

In any event, very few - if any - blacks probably arrived in the first colonies as truly free immigrants. If any did, they may have been former slaves or servants from other existing colonies. And even if they were indentured rather than enslaved, that does not necessarily mean that they went willingly.

Source material

• Figures taken from the Trans-Atlantic Slave Trade Database.
• Alden Vaughan, "The Origins Debate: Slavery and Racism in Seventeenth-Century Virginia", The Virginia Magazine of History and Biography 97, no. 3 (1989): 311 - 354. Though now quite old, Vaughan's article is a rigorous and valuable account of the major movements in the debate and key works in the scholarship and their points.
• Riva Berleant-Schiller, "Free Labor and the Economy in Seventeenth-Century Montserrat", The William and Mary Quarterly 46, no. 3 (1989): 539 - 564. Expounds on some of the methodological challenges to understanding black and white populations in early colonisation of the Americas.
• Hilary Beckles, White Servitude and Black Slavery in Barbados 1627 - 1715 (1990).

EDIT: Ridiculous typo.

EDIT 2: Just to say that I will be busy for the next couple of days and might not have time to get around to answering any follow-up questions, but I will try to when I have time!