Immigrants weakening the western economy?

Its not uncommon for people from less well-off countries to travel to the west to work in order to make more money. Some even work here, and send money back to thier native country to support thier families.

What effect does this sort of behaviour have on an economy? Does spending money youve earnt in your own country abroad weaken your home economy?

No.  The cheap labor immigrants provide boosts the individual wealth of the companies they work for.  "Suits" like owners, managers, and investors make a lot more money off immigrants, who work harder for less money, than they do from american workers.  The money immigrants send home is just a mitigating factor. 
What really weakens our economy is americans, rich and poor, spending money on goods made overseas where labor and pollution laws hardly exist.