The LocalSigner is not technically a part of the @apophis-sdk/cosmos-signers package, but instead built directly into the @apophis-sdk/cosmos package. It is a simple signer that can be used with a private key or mnemonic, and useful for CLI tools or server environments.
signers.ts
import { LocalSigner } from '@apophis-sdk/cosmos';

const privateKey = LocalSigner.generatePrivateKey();
const mnemonic = LocalSigner.generateMnemonic();

const signer1 = LocalSigner.fromPrivateKey(privateKey);
const signer2 = LocalSigner.fromMnemonic(mnemonic);
Once you have a LocalSigner, you can use it directly to sign transactions: 
import { Bank, Cosmos } from '@apophis-sdk/cosmos';
import { signer1, signer2 } from './signers.js';

const network = await Cosmos.getNetworkFromRegistry('neutrontestnet');

const tx = Cosmos.tx([
  new Bank.Send({
    fromAddress: signer1.address,
    toAddress: signer2.address,
    amount: [Cosmos.coin(1_000000, 'untrn')],
  }),
]);

await signer1.sign(network, tx);
await Cosmos.ws(network).expectTx(tx);
console.log(tx.hash, tx.status);