It's complicated.

The sender pays the fee, BUT the fee is dependent on the number of "inputs" (and "outputs") in the transaction. It's also dependent on the "traffic jams", sometimes (still!) transactions goes through almost for free, other times it costs a fortune. To complicate matters, the sender is free to choose how much fee to spend (and even 0 is a valid option) - but if the fee is too low, the transaction may not get confirmed, the receiver will not get the money. Even worse, the transaction may get "stuck" in the mempool for three days. Also, the receiver may try to spend "unconfirmed" funds, and if paying a sufficiently high fee, both transaction may confirm faster. This is called "Child Pays for Parent" (CPFP).

Let's take an example. I have received 5 BTC from someone, and I borrow 1 BTC to my aunt. It's a simple one-input-two-output-transaction; a 5 BTC input, a 1 BTC output and an almost 4 BTC output to myself ("change"). I pay 0.0001 in fees, so 1.0001 sent. The fee was too low, so my aunt will spend 0.001 btc when sending those funds further. In this case, the receiver actually paid most of the fee.

Then she pays me 0.001 BTC each day in interests. One day I have 0.1 BTC in my wallet, but spread over 100 inputs. There is peak traffic on the blockchain, and I need a fast confirmation. The transaction should have costed 0.001 BTC if it was a simple one-input-two-outputs-transaction, but instead it costs maybe 0.04, which is 40% of what I have in my wallet!

With SegWit, the input part of the transaction gets a discount. It won't help in the situation above, but if I empty my wallet, generate a new SegWit address and my aunt starts paying to that one, then next time I have 0.1 BTC in my wallet and need to move it during prime time, I will need to spend "only" 0.01 instead of 0.04.

With Lightning ... well, the problem is that I would need to spend probably 0.05 just to move those funds into the Lightning network, and yet more transaction fees to close the lightning channel ... and, more importantly, both the sender and the receiver needs to be connected to the lightning network, sharing some path (probably through some centralized hub) in order to use the lightning network. I tend to believe that still in a year, take two random peers, and the likelihood that they can easily transact through some shared alt-coin is higher than that they will be able to transact through the lightning network.