Heart rate monitors are accurate at measuring heart rate. Converting that into caloric expenditures is another matter entirely, and is often the result how accurate the equations they use to estimate the answer are. The key word is “estimate”…some are high, some are low. My best advice is to find the most conservative estimates (you’ll find a ton on the web) and use those if you’re really trying to lose weight.
As for cycling vs. running, it’s not all that straightforward. Spinning class (not the same thing as cycling) and running on a treadmill (not the same thing as running outside) further complicate matters. The #1 most relevant issue is how hard you’re working at each activity. Riding a bike in an intense manner CAN burn more calories over the same time as an easy jog.
In general, however, running is more efficient at burning calories than cycling in everyday practice. With running, you need to support your weight 100% of the time, more of your body is involved, and it’s impossible to “coast” like you could on a bike. On a bike, weight is an issue riding uphill, but not downhill, and most of your weight is supported by the bike. So if you have half an hour to exercise, most people will burn more calories running for half an hour than they could on a bike.
On a bike, terrain plays a much bigger factor in how hard you’re working. You may kill yourself going up a hill, but as you go down the other side, it’s common to reach speeds where pedaling no longer has any effect, and you end up coasting. Likewise a tailwind can give you a false impression that you’re working hard (if you use speed as a judge).
With running, you have no choice but to work hard. On a bike, you need discipline to work as hard (it helps to monitor your heart rate or power output so you don’t “cheat”) since it’s easy to trick yourself into thinking that you’re working hard.
Finally (and here comes the contradiction) if you want to burn maximum calories, get a bike. While running burns more calories over time (more efficient) riding a bike allows you to ride more TIME overall. So let’s say you’re able to burn 1000 calories/hour running, but only 500 per hour on the bike. Most “normal” runners that I know will run 5 miles or so on a given day, and maybe 10 miles for a “long” run unless they’re training for a marathon. That means that a “normal” run for less than an hour, and maybe up to a couple hours on a “long” run. Most cyclists, however, are able to for much longer durations due to the lower average intensity level of riding a bike. While a 45 minute, 5 mile run might burn 750 calories, a 3 hour, 50 mile bike ride might burn 1500 calories.