According to Colin McEvedy and Richard Jones, at the beginning of the Christian era it was about 2 million, and in the next 2 pairs of centuries it geometrically rose to about 2 2/3 million, then nearly 4. It had climbed to a high of about five and a quarter million around the time of Muhammad, a high-water mark that was receded from a bit to about 4.5 million within the first two Islamic centuries, not to be reached again until the Modern Era.
For this entire period, about half of this amount they place in The Yemen, the balance of the other half in the interior, and only trivial numbers on the Gulf coast and Oman. However, large amounts of the half in Yemen would not have spoken Arabic, but rather an assortment of related South Arabian (Semitic) languages.
For the region, these weren't gigantic numbers, but they were very respectable, and it was probably rather a lot for them. On population alone the Arabian peninsula was in the same league as the contemporaneous existing power centers in the boundaries of what today are Turkey and Persia*. McEvedy and Jones' text speculates that the high numbers likely put strain on the resources of this relatively poorly-resourced area, and that might partially explain its seemingly sudden foreign adventurism.
In Colin's excellent New Pengiun Atlas of Medieval History, he points out that the relatively easy victories the early Islamic armies had in the region may have been a reflection of the inhabitants not having a lot of fondness for the Greeks and Persians who had been fighting over their territory for centuries. Exploring this idea further, its worth noting that even if the locals didn't necessarily speak Arabic, Arabic is a Afroasiatic language, and such languages at the time were the language of the common people across the Levant and north Africa. So there was certainly at least some level of cultural affinity with Arabs that there would not have been with Greeks and Persians (both speaking Indo-European languages).
Pre-Islamic distribution of Afroasiatic language families
Image taken from Expansion of Afroasiatic Languages video on Youtube, based "loosely" on the work of Harold Flemming.
Distribution of Semitic Languages
* - The boundaries of modern Turkey at that time they put at around 6 million, and Iran at about 5.