tl;dr We don't really know. A reasonable guess is that it's a weak sink, gradually turning into a source via increased drought frequency, but it's extremely hard to measure; carbon flux is highly variable both spatially and temporally (so it matters both when and where you measure), and it's difficult to make sure that you're accurately measuring all the different components of carbon flux.
Dolman and Janssen 2018 say:
Ground observations suggest that, through tree growth, the Amazon forest acts as a net carbon sink to the atmosphere (Brienen et al 2015), which is confirmed by atmospheric studies (Gatti et al 2015). However, the sink strength appears to have been declining since the 1990s and recent drought events in 2005, 2010 and 2015 might have temporarily reversed the Amazon carbon sink into a source (Feldpausch et al 2016, Gatti et al 2015)
They go on to discuss the great difficulties in constructing an accurate model of Amazonian carbon dynamics (for the purpose of predicting future source/sink relationships).
Rice et al. 2004
summary based on measuring living trees and coarse woody debris [CWD], it looks like a particular patch of forest was a small net source, but this may be a transient effect.
Amazon forests could be globally significant sinks or sources for atmospheric carbon dioxide, but carbon balance of these forests remains poorly quantified.
In other words, "it's still unclear [as of 2004], so we did another study."
We surveyed 19.75 ha along four 1‐km transects of well‐drained old‐growth upland forest in the Tapajós National Forest near Santarém, Pará, Brazil (2°51′ S, 54°58′ W) in order to assess carbon pool sizes, fluxes, and climatic controls on carbon balance.
So they took a very small sample of a particular forest type.
The gain in live wood biomass was exceeded by respiration losses from CWD [coarse woody debris], resulting in an overall estimated net loss from total aboveground biomass ...
So this particular patch of forest was a carbon source at this particular time ...
... a period of high mortality [probably] preceded the initiation of this study ... transfer of carbon between live and dead biomass pools appears to have led to substantial increases in the pool of CWD, causing the observed net carbon release.
... but that may be due mostly to a recent die-off - maybe not true over a longer time scale ...
... the hypothesized sequestration flux from CO2 fertilization ... would be comparatively small and masked for considerable periods by climate‐driven shifts in forest structure and associated carbon balance in tropical forests.
... it will probably bounce around a lot (i.e., it's not wise to extrapolate this result either in space or time).
Miller et al. 2004
We used two independent approaches, biometry and micrometeorology, to determine the net ecosystem production (NEP) of an old growth forest in Pará, Brazil. ... Biometric inventories indicated that the forest was either a source or, at most, a modest sink of carbon from 1984 to 2000 ... biometric and micrometeorological measurements in tandem provide strong evidence that the forest was not a strong, persistent carbon sink during the study interval.
and a more recent reference ...
Antonucci et al. 2018
The objective this paper was to quantify the hourly variability of CO2 fluxes in the year of the El Niño, in 2015, in a tropical rain forest in the Western Amazon ... The average daily behavior of CO2 fluxes showed higher concentrations of absorption during the day than emissions overnight, behaving as a carbon sink, corroborating the importance of the forest in assimilation of the atmospheric carbon.