Whether or not a public school will restrict entry to local students depends on individual schools,
For Catholic schools however, yes there generally are waiting lists, particularly for Secondary schools, but some have longer lists than others. Your friend is referring to some Catholic/Private schools where demand is so ridiculously high that parents are signing children up at birth. This is only really essential if a) you don't have prior links to a school, b) you live out of the catchment area and/or you're not practising the school's religion, c) you really don't want to see your child going to school anywhere else.
However, there are still plenty of alternatives, at the end of the day it comes down to the child as well.
1 'Private'/'Catholic' vs 'Public' Primary Schools
The main reason for sending your child to a 'Catholic' (or some sort of Private/religious school) is if you want them to grow up learning a certain set of (religious) values, &/or if you are of a certain religion (in this case 'Catholic') and you want/need your child to go through the sacraments. That is not to say that 'Public' schools are 'values neutral', and there'd still be some sort of religious 'Scripture', albeit at a minimum.
2. Academically, at Primary school level, there are fewer points of difference except if you hope to send your child to an academically selective public high school, you may have some difficulty if they're in the Catholic system, as they tend to prefer to keep kids 'in the system' and send them to other Catholic schools (I speak from experience). I'd imagine most primary schools would try to cater for gifted students, however, designated public schools have 'Opportunity Classes' that require an entrance test and specifically cater for this (in the Rockdale area, this is offered at Hurstville Public).
Aside from this,you should also consider school/class sizes - are you happy sending your child to a large school with multiple and composite classes in any one grade, or do you prefer a small school where your child is one the one small, tight-knit class from K-6 (this was probably the standout factor of my primary education at a Catholic School). Small schools/classes can be found in both private/public. Specifically in the Rockdale area the Catholic schools generally only have 1-2 classes, and some 'small' public schools include Bardwell Park Infants (K-2), and Kyeemagh Infants, but these are some way from Rockdale too. In the City, I think Fort st Primary, and schools in Surry Hills also have small numbers.
This of course, depends on other factors as well - certain programs, facilities, and of course, in your case, before/after school care. Most schools would try to have some arrangments, but some have on-site before/after school care.
There's so much information you can access, here are the two main websites
In looking at schools, you may also want to look up locations on the map to see proximity to train lines, and look up bus routes as well for accesibility if public transport will be an issue.