Bitcoin is meant to be a decentralized network not requiring trust to make/receive transactions. Transactions contain signatures to prove that they haven't been tampered with. Signature is not just the private key, it is something that is derived by the private key and some other parameters. This is called ECDSA, Elliptic Curve Digital Signature Algorithm. If the crypto implementation is proper, it's impossible to extract the private key from signatures.
Changing the blockchain is a different topic. Each block contains "Proof Of Work", which means its hash (a value to check a block's authenticity, but no private key) would change if any part of the block is altered. This'd invalidate the block, and miners would have to re-mine that block (change little things until the hash is lower than a value, the target) But if you create a fake chain with lower targets for blocks, then your chain would have have less Proof Of Work accumulated and other nodes would switch to the chain with the largest PoW once they're aware of it. (But in practice there are checkpoint blocks, so the initial target for remining a chain is high)