Ignore my earlier comment. I found the solution on the org.apache.http logging page. Your original answer was referring to httpclient-3.x logging, and the working code for recent versions comes from http-components logging
java.util.logging.Logger.getLogger("org.apache.http.wire").setLevel(java.util.logging.Level.FINEST);
java.util.logging.Logger.getLogger("org.apache.http.headers").setLevel(java.util.logging.Level.FINEST);
System.setProperty("org.apache.commons.logging.Log", "org.apache.commons.logging.impl.SimpleLog");
System.setProperty("org.apache.commons.logging.simplelog.showdatetime", "true");
System.setProperty("org.apache.commons.logging.simplelog.log.httpclient.wire", "debug");
System.setProperty("org.apache.commons.logging.simplelog.log.org.apache.http", "debug");
System.setProperty("org.apache.commons.logging.simplelog.log.org.apache.http.headers", "debug");
and properties:
adb shell setprop log.tag.org.apache.http VERBOSE
adb shell setprop log.tag.org.apache.http.wire VERBOSE
adb shell setprop log.tag.org.apache.http.headers VERBOSE
The difference is in the logging tag names. Reference: