I’m not asking about how European disease spread. It’s is clear that it spread along trade routes through trade routes and other routine tribal national communication e.g. marriages, shared borders, meetings of leaders.
Settlement was limited in the first few decades after European arrival and was relatively scattered. For example the British settlement of North America was limited to groups of settlements along the Atlantic coast of North America. Spanish expeditions went further into the interior, but were still not comprehensive and not necessarily for demographic data gathering.
Demographic data from that era would have to have a reasonable range: e.g 80 million with a margin for error of +- 5%. However, how does a demographic historian reach an accurate number for population groups in an area as vast as the America, many decentralized population groups, and probably a majority of population groups that did not record their own demographic data.
It's a very good question that everyone needs to ask before embracing one figure or another for periods for which any particular estimate may be insecurely-founded, speculative or legitimately contentious.
The first thing to be said is that our demographic aggregates for the period before 1500 or in most places even 1700 are unlikely to come with so little as a a 5% error margin anywhere, and that includes Europe or China for which we have a range of contemporary records of taxpayers, households or sometimes reported persons. For relatively well-documented England with its 1377 tax count, estimates for the eve of the Black Death range from 3.7 million to six million.
The problem is inevitably magnified greatly in the case of the Americas where no precolumbian count of any regional population or even a large part of it survives. Conventional rules-of-thumb like multiplying plausible household returns by five or whatever are frankly useless: there's just no contemporary return unless we can be fairly confident that a later estimate is consistent with a community's likely size before any significant adverse contact impact.
That's why we have estimates ranging from Alfred Kroeber's 8.4 million (no more than at the 17th-century low, if that) to Henry Dobyns's 90-112 million (assuming uniform decline of 95% or 96% to each regional low) . For North America alone, the proposed numbers range from 900,000 to 10-12 million. While the very low counts have lost ground in the last half-century or so, the high-counters have also come in for criticism, and there remains little in the way of a consesnsus. So where do the numbers come from? "Nowhere", answered David Henige (1998), but the two main approaches might be summed up as estimating "carrying capacity" (how many people a territory might be expected to support given the available technology, land productivity and intensity of use - and back-projection based on population change overall or in a given interval: each has yielded differing results, the former tending to produce lower estimates than the latter.
The ascendancy of the "high count" was triggered in the 1960s by the "Berkeley school" estimates of Cook, Borah and SImpson who estimated the population of central Mexico alone at 25.2m in 1519, applying multipliers or rates of decrease at successive intervals following the Spanish conquest, though their work was critiqued robustly by Zambardino (whose upper bound approached Cook's 1948 estimate of 11m) and Henige (who concludes that there is no adequate basis for any total). Some of the figures have tended downward from recent levels, though not to the very low numbers of early researchers: For North America, detailed analysis by Ubelaker and Snow based on cultural or ecological areas has indicated a pre-contact population of 2-3½ million - higher than the prewar 1m but appreciably lower than Dobyns's revision.
There is no simple answer: for now we have widely differing accounts based on very different assumptions, with little common ground. For my part, I find neither the low nor high counts persuasive, but think the Ubelaker/Snow approach may be more productive: that might suggest a precolumbian total of the order of 30m, or applying it across the hemisphere might produce something quite different. But it's best to be wary of any one number.
For the "high count", see Sherburne Cook & Woodrow Borah, Essays in Population History: Mexico and the Caribbean, University of California Press 1971 and Henry Dobyns, Estimating Aboriginal American population: an appraisal of techniques with a new Hemispheric estimate, Current Anthropology 7:4 (Sep 1966); their critics include Rudolph Zambardino, Mexico's population in the sixteenth century: demographic anomaly or mathematical illusion? Journal of Interdisciplinary History 11:1 (summer 1980), and David Henige, Numbers from nowhere: the American Indian contact population debate, 1998. . For North America, see Douglas Ubelaker, Prehistoric New World population size: historical review and current appraisal of North American estimates, American Journal of Physical Anthropology 45:3 (Nov 1976), and North American Indian population size, AD 1500 to 1985, American Journal of Physical Anthropology 77:3 (Nov 1988), and Dean Snow, Setting Demographic Limits, The North American Case, in Computing Archaeology for understanding the past, Ljubljana, April 2000.