http://www.knifeup.com/oregon-knife-laws/
This is Oregon. Most places in Oregon an axe is considered a tool.
Transporting a sword or axe is not necessarily a big deal unless you go to hot spots like schools, banks, federal buildings, etc.
Now, if you are doing something weird with either of those you WILL draw attention and potentially be talked to. Busy body neighbors and people who are afraid of anything out of the ordinary might make a call to LEO.
I posted awhile back about buying a double bit at a flea market, carrying it back to my truck and everyone looked freaked out- especially a bearded guy in a flannel shirt that from all outward appearances looked like a throwback lumberjack - that was Portland.
If you are younger, you might draw attention based on an "up to no good" assumption. If you two decide to have a chop off somewhere outside of your own property near a larger community you might get looks.
Outside of not carrying either in a ready position I don't think there is a legal issue in transporting them.
The community I grew up no one would have thought twice. Just be smart and respectful of property and people and the only problem will be someone else's.
Now, if I got talked to by the cops my dad would have been my biggest fear...
Things are so "dangerous" in today's society I'm surprised any of us made it to adulthood.