When I do dungeons I feel that the more paladins the better. The more blessings the better, specially for the tank who can get wisdom + one more, or two more, the more the better.
Prot paladins are super good for dungeons up to lvl 60 as they actually do damage compared to warriors (spend the first 11 point in holy to get concecration). You will for example see a prot paladin do 300dps when the warrior do like 50dps... Paladins have waaaaaaaaaaaaay better aoe damage/threat also.
The only real problem with a prot paladin is when for example a mage sheep a target. When it's time to attack that target the healer will have a lot of aggro from heals, so the sheeped target will run for healer and the tank will struggle to get aggro.
That's why it's good to have 2 paladins, as you can keep Blessing Sancturay on all players (30% less threat) + one more blessing so to have a paladin tank and paladin healer is really good. I can't see any negativ thing with that + you can use bubble if some mob or boss attack you and tank just can't get aggro back.
I am sure druids are fun to, but I know for a fact, if you and your friends play as paladins you can have a great time

I don't see how you would compete for the same gear. As healer you go for +healing and +int (often cloth/leather). The prot will go for mail/plate stamina, strength and defense etc. Maybe you will both need the same shield at some point but that's unlikely.
Personally I love the paladin healer. You don't have any aoe heals but you have some other awesome tools to use.