You start by determining the heading, you can do that by using the Google maps geometry library, it has a function called computeHeading() you pass in 2 coordinates and it'll return a numeric value of the heading of the 2 coordinates.
Here I'll explain how to interpret the heading to N, E, S & W.
Lets assume we have a clock, 12:00 is a -0 or a 0 heading which on the compass is North. 3:00 is heading 89.5 which on the compass is East. 6:00 is heading 179 or -179 which on the compass is South. 9:00 is heading -89.5 which on the compass is West.
When you get the heading returned from the function. if it's in the range of -44.75 to 44.75 the heading is North, If it's in the range of 44.75 to 134.25 you can determine the heading is East, and so on.
Your question is regarding the side of street the car is on, that's almost impossible to figure out without any user input (the user selects if they're on the right or left side of the street) since the car might not even be parallel to the road.
The heading does help you determine the direction the car is facing and based on the country the car is in you can presume it's driving on the side of the road driven in that country.