This really depends on what browser you are using. For Mozilla and Fire$ANIMAL, there is liveheaders, which displays the sent headers in a separate window. If you are using another browser, you might want a proxy which dumps all traffic going through it. Such a proxy could be written with HTTP::Proxy for example, but other solutions are also possible, by simply sniffing all traffic with ethereal or Net::PCap.

Even if you are using WWW::Mechanize, it's not a panacea - there will come a time when you really need to see what the differences are to make WWW::Mechanize mimic a specific browser in the closest possible way.