People have been saying this kind of thing to me ever since I ran the PC department at British Airways, and my answer has always been the same. If all you are doing is handling lightly formatted text, cheap and cheerful is fine, but as soon as you use the more sophisticated aspects of a word processing program, this kind of transfer becomes risky, and simply isn't worth it.
I've just had a good example of how things can go wrong using OpenOffice. I was sent a document to check as an ODT file - the file format from OpenOffice. It had a series of appended comments. The file doesn't open in Word or Pages, but I tried it in both Google Documents and Textedit and neither showed the comments. No problem - the person who produced it exported a Word document from OpenOffice for me. And, yes, it did have comments - but they had been bizarrely scrambled. The image above are some of the actual comments, rendered utterly unreadable - and none of them pointed to the right bit of body text. It was garbage, pure and simple.
In the end, I had to download a copy of OpenOffice and work on the original with that. As it happens this was fine, because it was this way round - OpenOffice happened to be the standard used by the company I was doing some work for. But almost all publishers, magazines etc. expect material in Word format. And if you are working in OpenOffice, you will have to export your document to Word. Potentially with the kind of result I just experienced.
By all means use OpenOffice for printed documents, or those for internal use. But if you intend to share anything more sophisticated that straightforward text with a publisher, say, in a professional capacity, then think twice about turning up your nose at Word.