In New Zealand where we don't have a specific constitution or amendments we have a set a laws ^1 that end up in the same place. An example is libel, which both countries have laws against. In NZ such laws are debated in parliament and voted on just as in the US. However in the US there was an additional objection based on it violating the first amendment but then the law was made anyway so it seems politicians in the US can make laws that override amendments in specific situations. The US also has their Supreme Court which seems to play a far more active role than NZ's and also more powerful in that it can creates precedents in the interpretation of laws for example allowing students to wear items of symbolic protest in school.

^1 In 1990 we got a law called the Bill of Rights Act which included freedom of expression.

Edit: added ^1