To NWBound: no, not naive..
If you mean "investing IN the European market", at the time I felt I was already overweight in int'l. stocks (>50-60%, back when that was very racy). Buying & holding and fully invested, using the cash to build up more in (supposed) US "blue chips" since that's where my allocation was weak at the time.
If "investing ON the European market" you mean with an account overseas there were a number of factors:
1.) Pollyanna (ducking) prejudice: "the US is strong; the US is best; our economy can't be beat; we are more transparent, etc." I drank that Kool-Aid at the time.
2.) having one foot in each world and wanting not to burn all my bridges. I fell down the rabbit hole into Italy and wanted the rabbit hole there to get back up and out to the 'real' world. But things got inverted and now the US seems wackier than here.
3.) convenience factor, dealing with institutions that 'make sense' to me. In 2000-2002 when we were planning this move and trying to figure our lives out, Italy didn't have accessible online brokerages, or really any pure brokerages that I know of. Investing is done through banks, and the banks are not that customer-friendly. They love to put people in (wait for it..) annuities and oddball equities mixstruments that guarantee a maximum/minimum linked to the performance of some basket of stocks (kind of like a hybrid between an annuity, a managed fund, and an ETF). I wasn't desirous of having to physically go to the bank and wrangle with a rep to make individual stock trades. Only a tiny minority of Italians own any stocks at all (a couple years ago I heard 5%) so there's not been the experience or the commodification of trading aspect on the part of institutions.
4.) Expense. Banks here have also traditionally had ridiculously high fees, even fees to close the account. You could leave money in savings, forget about it, come back later and find an invoice for fees in arrears because the fees ate all your money. This has changed somewhat lately with reforms, and now there are a few American-style online brokerage/bank operations with inexpensive trades and low fees (or waived fees with a higher balance). This has all developed just in the last couple of years.
5.) Ah! and don't forget, the euro was brand spanking new in 1999-2000. What was its track record? The monetary union could cede at any time. Everyone was skeptical, including the europeans, about its future. The Italians were furious at the euro inflation and the Second Coming of Berlusca in 2001 was threatening euro stability on a couple of levels. Who knew that the tables would be turned in such a short time?
6) House purchase. Sold my US house owned outright and put almost all the value of that into a house here, making one big cash transfer. There was an amount left over to live on for a year, year and a half as well, before I had to start withdrawing from the US. The house purchase at the time looked to me to be a whopping big euro investment (in fact it was 1/3 of our egg).
7) I couldn't really have foreseen that • the US would go to war and that • people would go SO much into debt to buy houses.
I turn my back for ONE minute and you guys break the country..! WTF! 
8.) In the interim, no worries. I call Schwab; they send money. In 3/4/5 doses per year, so I figured I was kind of DCA'ing as I went along and investments doing ok so as long as I had more dollars I wasn't as concerned as I should have been that they were buying less... until I started hearing more about the housing crisis, and started to investigate more what is behind it. Now my fear for the dollar is pretty serious.
I'm not sure what would have been better.. IF I had been watching continuously then maybe I could have "done something" and adjusted??.. yet now many recommend I should just take a chill pill. But I'm the one responsible for the money and I feel I would let DH down if I can't avert collision with a possible oncoming train.
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In hindsight, I would have done two things differently. This is not 20/20-type hindsight on my particular situation, just my biggest recommendations for retirees looking to live in another country.
a.) buy an "average" abode. I was tempted to buy a house about 80% of the then-dollar value of the one I left behind, without taking into consideration the local std. of living. It's a very nice stone house, with a big yard. It's huge by Italian standards, and small by McMansion standards.. 4BR, 3 bath. The problem was we wanted a detached house, not an apt. condo, and those are few and far between on the market here. Anyway your wallet will thank you, you will fit in better with the locals, and you will have lower expenses in taxes, heating (ouch!, it's our biggest expense after food) and maintenance.
b.) make sure you have a firm timeline. Think hard and decide if this is a move for 10 years only, for 20, or for forever. Then try to adjust your investments and currencies accordingly AND adjust your mindset. We were just 'winging it' and have always been figuring "well, we
could go back to the US, so..." but that is really deadly on more than one level: it's psychologically problematic because you are not fully invested emotionally in the new surroundings - they are 'temporary' and so you don't always devote attention to the right things, and financially because you could be blindsided like me. Of course if you have a really whopping amount of money you can do as you like.. I'm just talking about a modest retirement. I have heard of a couple of expats who retired on fixed USD pensions and have been forced back to the US due to exchange rate woes; you don't want to be one of those statistics.
