Do east side and west side implicate poor and rich respectively?

Solution 1:

Usage of the terms started in London and other industrial cities after the industrial revolution.

The sudden increase in the burning of coal for both industrial and domestic use created a huge pollution problem. As the prevailing winds in the temperate zones are from the west, the smoke blew to the eastern parts of the cities. Within a few decades the eastern parts of cities were dirtier and more polluted than the west.

Very quickly, those with money moved to the West End and those without were left in the poor accommodation of the East End.

In turn, the terms East End and West End became associated with social standing and, by extension, crime rate.

A description of the distribution of "The Great Stink" in London is given by the Museum of London.

The problem is by no means unique to London. At the time of the industrial revolution, most cities with such problems were in areas with prevailing westerly winds.

Solution 2:

These "sides" are referring to the east side or west side of specific cities. These terms usage are therefore not able to be used with all cities. However, LA is a very popular place for musicians to come from and reference; both songs you mentioned are most likely referencing LA, which has a specific area called Eastside.

Another common "Direction"-named neighborhood is Chicago's South Side.

These neighborhoods have become ubiquitous in media that it is sometimes no longer necessary to reference the city, but just use the neighborhood.

To give a counter-example the city that I am from, Cincinnati, the West side is typically viewed as less wealthy than the East side.

Typically these words are used just as any other geographic region or area would be used.