Actually, all hosts on a network do claim their own IP address themselves. Always!
On the local network side a router has an IP address in a certain range (e.g.
192.168.9.x) and if a router receives a packet of bytes for, say, 192.168.9.12,
the router forwards it to its physical port that has the IP range
(192.168.9.0/24) configured that matches the destination address. On that port, it broadcasts on the local network: "Which host
claim IP address 192.168.9.12?". And any host that has its NIC configured to
have that specific IP address will answer, thus claiming the address. If all is
well, there is exactly only one host that claims it, and the router will send
the data packet to that host.
This broadcasting "who has IP address x.x.x.x" and claiming it by replying,
happens at a lower level and only works on a LAN (local network). This protocol
is called "ARP" (Address Resolution Protocol)
To put it in other words:
The address the router sends the data package to, on this lower level, is
actually the MAC address of the network interface card (NIC) of the host that
claims the IP address. It does so by replying to the ARP-request broadcast by the router.
So, it really is the host itself that decides what IP address it will
claim to have on a network.
On networks bigger than just a few hosts it will be a tedious task to keep
track of what IP address is to be used by what host. And then imagine editing a file on all computers allowed on the network to only tell it its IP address!
Also, if two (or more) hosts claim the same IP address, both will have a
serious network problem! That is why it is very practical to have a central
service on the local network that keeps track of which MAC address can use
which IP address. This is called "DHCP" (Dynamic Host Configuration Protocol), which can also be used to inform hosts
of many other configuration parameters on the local network.
The DHCP protocol also operates on the same lower level as the ARP protocol. When a hosts
configures its NIC's (Network Interface Card), most often at boot time, it
broadcasts "Which IP address can I use on this network?". This is done by DHCP
client software. Since it's a broadcast, all hosts receive it, but only a DHCP
server will answer, hopefully mentioning a unique IP address, its netmask, the
gateway (router) address and possibly other things. The host keeps this in
memory knowing it can safely claim this IP address on the network for some time.
But this only works because the hosts cooperate by asking a DHCP-server for an IP address to use. Ultimately it is always the hosts who decide what IP address they'll claim....