Make photocopies of your passports, including the page with your photo on it. Take a set of copies with you (I hang on to my wife's copy and she holds mine, a nice bonding thing to do and good advice based on the premise that if I lose my stuff it will be good for her to have the copy). Leave a set of passport copies at home with someone you trust and who won't mind you ringing up at 2 am their time so you can ask them to FAX the copies to Papua New Guinea.
If you lose your passport abroad, you will have to get it replaced at an American Embassy or Consulate in the host country. The photocopies will help but may not be all you need. If you lose your passports, as soon as possible, call the American Embassy and find out what you'll need to get a replacement. If you have three days left in Belgium and decide to see a few sites, it will be useful to know the Embassy will need two days to replace the passport before you drop in on the way to the airport thinking of using a drive through window (they actually don't have drive through windows, that was a bad joke).
Embassies are generally open “business hours” which means you will not get a replacement passport absent a real life or death emergency after hours. If you are replacing a child's lost passport, ask when you call the Embassy if the child must appear in person or if one parent can do the paperwork. You might be able to draw straws with your spouse and have only one of you miss the Paris Sewers tour while the other and the kids go forth underground.
The Paris sewer tour is excellent, allowing you to wander around under Paris' most famous addresses. Extra fun for Phantom of the Opera or Ed Norton fans. It does not stink down there like poop either. Think wet sweater smell.
Most Embassies and Consulates do not have photo booths where you can get the pictures needed to replace your lost passport. The best idea is to get a few extra photos made when you take the pictures for your first passport back at home and carry them along. If all else fails, you can always spend more vacation time abroad enjoying the company of a photo developer in Rome.
While we are on the subject, the Department of State has a lot of information that will help you with your trip. One thing I'd recommend is their booklet A Safe Trip Abroad. You should be able to get it on-line, at some larger passport agencies or if all else fails, from the trusty Government Printing Office. Write to them at Government Printing Office, Superintendent of Documents, Mail Stop SSOP, Washington, DC, 20402-9328. The ISBN Number for this booklet is 0-16-048791-9.
You can also get info sheets and pamphlets by mail by writing to Overseas Citizens Services, Department of State, Room 4811, Washington DC, 20520-4818.