Detecting if point is on left or right side of line in PostGIS?

select (ST_Azimuth(h.vec) - ST_Azimuth(h.seg))
from (
    select 
        ST_MakeLine(cp.p, point.geom) vec,
        ST_MakeLine(cp.p, 
            ST_LineInterpolatePoint(
                line.geom, 
                ST_LineLocatePoint(line.geom, cp.p) * 1.01)
        ) seg
        from (
            select 
                ST_ClosestPoint(line.geom, point.geom)
        ) p as cp
    ) as h

So the idea is to calculate angle between closest line segment, and vector from closest point on the line to your point.

get a closest point on a line

select ST_ClosestPoint(line.geom, point.geom)

create the vector from closest point to your point

ST_MakeLine(cp.p, point.geom) vec

create a vector among your line

ST_MakeLine(
    --original point
    cp.p, 
    --find a point next to the closest point on line
    ST_LineInterpolatePoint(line.geom, 
         ST_LineLocatePoint(line.geom, cp.p) * 1.01)) seg

get the difference between directions

ST_Azimuth(h.vec) - ST_Azimuth(h.seg)

So right and left will be greater than zero and lower than zero.

An alternative method if you are using version 2.5.0 or later of PostGIS, you can supply a "side" argument to ST_Buffer. So an alternative method to the ST_LineInterpolatePoint with an scaled projection value is to use

ST_Contains(ST_Buffer(line.geom, 100, 'side=left'), point.geom)

One downside of this is the need to give a distance(in this case 100) to build the buffer on the left side of the LineString but in my case I already have a threshold of how close point.geom needs to be to be associated with line.geom.