Jammy, I have a 16mo and am just about to have another lo.
My dd can throw the best tantrums, I just leave her where she throws herself and let her get it out of her system for a bit. Stuff what anyone else thinks around you it's your child and none of their business as long as your ds is safe.
There are various tactics for dealing with tantrums:
Have a think about what sets him off. A lot of children get ratty when tired or hungry so maybe avoid going out when it's close to sleep or meal times, or take snacks and drink with you.
If you can see one starting to brew you could try creating a diversion (eg. I just saw a monkey over there, can you see it), but this takes a lot of imagination and effort on your part which is not always possible with another baby in tow.
I would say don't back down and start giving in to his demands, this is a time in his development when he's testing limits and if he knows he'll get what he wants with a tantrum then it won't give him an incentive to stop. Just keep telling him you love him when he's screaming (you'll probably need to remind yourself that you do!) but explain firmly why he can't have his own way.
Having said that, if it's really not a battle that you need to fight then relax and go with what will ease the situation.
Don't be frightened of tantrums, they are a necessary part of growing up and won't do him any harm at all. Being 2 is a very confusing time. Just keep him safe and loved and it will pass - and remember, don't give a stuff about what other people think.