1) When I Google "double size postcard", I get many different results. For example, see all the options in the following list of standard postcard sizes:
http://www.uprinting.com/standard-postcard-size.html
I realize that you used this terminology just as an example, but in the future, and to make this discussion concrete, can you simply give us the dimensions IN INCHES of the largest image that is to be printed (...on the postcard or however else it will be printed...).
While we are awaiting that info from you, let's assume that the largest image to be printed is 10 inches (on the long side), a nice round number to work with.
For reasonable quality prints, you should have at least 200 pixels per inch (ppi), and preferably, 300 ppi. This means that the pixel dimensions for such an image should be at least 2000 pixels ( = 200 ppi X 10 inches), and preferably 3000 pixels in the long dimension.
However, both of the examples you gave were well under this. So, you can see why I need you to be specific in terms of image size IN INCHES (or standard papers sizes as listed here:
https://en.wikipedia.org/wiki/Paper_size ).
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2) With respect to your suggestion to down-rez images before transmitting, just don't do it. It's a complete anachronism. These days,ordinary people have devices (eg, cheap point-and-shoot cameras, cell phones, etc.) that routinely produce images that are 3000 or more pixels on a side. They also email them routinely. If your friend's email account can't handle an attachment of that size (ie, typical file sizes are several megabytes), then just send it to him using one of the free file sharing services like YouSendIt.com, Dropbox or any of the equivalent cloud based offerings from Google, Microsoft, etc.
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3) With respect to your concern about "seeing individual pixels", that's not going to be a problem for any commercial printing service, and not even for most home inkjet printers. This is because even if you send the printer an undersized file, the printer driver (ie, its software) will automatically up-rez your image before actually delivering it to the printer. If it up-rez'es your image, the worst you will see is a kinda-lumpy, slightly blurry image, but almost never will you see individual pixels (except in some very odd cases that aren't even worth talking about).
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4) You spoke about making relatively minor changes in the pixel dimensions of your images (eg, from ~1300 px down to about ~1100 px). Don't do this either. First of all, for the reasons I stated in #1 and #2, above, you don't need to, and you shouldn't make your image any smaller than it already is, and secondly, every time you resample an image (ie, change the pixel dimensions without cropping), you lose information and introduce a slight amount of blur. This is why Gedstar responded in the way he did.
HTH,
Tom M