No - Your coins are safe under the single private key (i.e. wallet) on both chains, as all transactions prior to the fork are safely recorded on the blockchain for both forks (assuming the transaction out to the paper wallet was recorded to the existing single blockchain before the fork).
Your private key can be imported from the paper wallet to a wallet for both branches of the fork, meaning you'll have two wallets with the same amount of coin on each of the two blockchains - these can be sent independently of each other to wherever you like (as each branch of the fork will ignore the transactions of the other - in theory).
The only reason to commit to one of the two would be the perceived value of one over the other, and whether you want 'double' your eggs in one basket (perceived financial value ratio of one to the other notwidthstanding).
Again, the value of both will more than likely take a hit during (and for a while after) the fork, so please don't harbour illusions about doubling your investment overnight!
There will be plenty here on HN who know more about this however, so any corrections to the above are welcome.
Your private key can be imported from the paper wallet to a wallet for both branches of the fork, meaning you'll have two wallets with the same amount of coin on each of the two blockchains - these can be sent independently of each other to wherever you like (as each branch of the fork will ignore the transactions of the other - in theory).
The only reason to commit to one of the two would be the perceived value of one over the other, and whether you want 'double' your eggs in one basket (perceived financial value ratio of one to the other notwidthstanding).
Again, the value of both will more than likely take a hit during (and for a while after) the fork, so please don't harbour illusions about doubling your investment overnight!
There will be plenty here on HN who know more about this however, so any corrections to the above are welcome.