In order to apply for permanent residency in Paraguay, I have to get some documents "legalized".
What does that mean, and how do I do it?
I'm currently located in the US, and I'm a US citizen, if that is important.
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Legalization is a process by which a consular agent validates the legal status of a document. See Wikipedia for a full explanation.
These instructions cover the process to legalize documents (e.g., for your permanent residency application) at the Paraguayan consulate in Los Angeles. The process might be slightly different depending on which consulate you need to go to.
You can visit the nearest consulate to you to have your documents legalized. Some countries require that you submit documents to the consulate that has "jurisdiction" over the state where the document was issued, but I was able to get documents legalized just fine at the LA consulate despite those documents being issued from multiple states across the US.
The Paraguayan consulate in LA is located in a big office building at 9841 Airport Boulevard, Suite 820. From LAX, it's about a $7-10 cab ride. If you are driving your own vehicle, note that parking is not free (but it's not terribly expensive; if you are in and out within 15-30 minutes each visit, it will cost you less than $5 each time, and the parking garage attendant accepts credit cards).
The consulate publishes the legalization price list online. As a general rule, expect each document to cost between $30-35.
This is not a same-day service. After you drop off your documents, it will take a couple of days before the legalization is done.
Be sure to mention that you are from out of town and visiting specifically to get your documents legalized. For some reason, if the receptionist thinks you are local, she will quote a significantly increased turnaround time. In my case, she originally said it would take 5 days, but when I mentioned that I had flown in from Texas, she told me to try back the next day (although they weren't ready until the following day).
Also be aware that your documents might not be ready when they say they'll be ready. When I went to get my docs legalized, the receptionist told me to call back the next day to see if they were ready. I stopped by the next day, and she informed me that the person who does legalizations wasn't even in that day!
Keep in mind that you are dealing with bureaucrats, and also from what I've heard, it is part of Paraguayan culture to tell white lies to save face.  Be patient, check back frequently (I recommend showing up in person every day), always be polite and friendly towards the receptionist, and you will eventually walk out of there with your documents.
What does a legalized document look like, you may ask? There's two components to the process:
Don't expect the end result to look pretty. When I got my docs legalized, they ripped off the apostille on my FBI background check, and the stamp had so much ink in it that it bled through to the front of each document. I checked with my fixer, and she said that's OK.
I am not an attorney, tax professional nor financial advisor. Nothing contained in this post should be considered legal, financial, medical or any other kind of advice, for that matter. Always do your own research and due diligence before getting involved with foreign governments and offshore service providers.