I'm going to keep it short, because there's a lot of technical background.
So, I'm assuming you know about:

So, to business:

Intro

Event viewer data is stored in .evt/.evtx files in %SystemRoot%\System32\Winevt\Logs\, but as with any half-decent DB, no one but SYSTEM is touching the files themselves.
The permissions are basically divided into two interesting parts - event-related permissions (reading / writing / clearing event logs) and source-related permissions (creating / deleting sources).

Other permissions

Of course there is also creating / deleting logs themselves, but it happens so rarely that I just do it with my admin account.

Event-Related

Event-related permissions are per log (application/system etc.) and are stored in an SDDL in HKLM\SYSTEM\CurrentControlSet\services\eventlog\LOGNAME:CustomSD
To grant or deny principles some privileges, edit the SDDL, using the following bits:

  • 0x01: Read
  • 0x02: Write
  • 0x04: Clear

Note: Restart is required to apply those permissions

Scripting Time

I use PowerShell to read the registry value, parse the SDDL into an ACL object, edit it, and export the SDDL-equivalent back into the registry.
This stub of a script will grant some principle permissions to read from and write to a certain log:

param(
[string]$Principle = 'Builtin\Remote Desktop Users',
[string]$LogName = 'Application'
)

# Get SDDL:
$orgSDDL = gp ('HKLM:\SYSTEM\CurrentControlSet\Services\Eventlog\'+$LogName) CustomSD | select -exp CustomSD;

write-host 'Before:'
$orgSDDL;

# Create ACL
$acl = new-object System.Security.AccessControl.RegistrySecurity;
$acl.SetSecurityDescriptorSddlForm($orgSDDL);

# Create ACE
$ace = New-Object System.Security.AccessControl.RegistryAccessRule $Principle,3,'allow';

# Combine ACL
$acl.AddAccessRule($ace);
$newSDDL = $acl.SDDL;

write-host 'After:'
$newSDDL;

# Store SDDL:
sp ('HKLM:\SYSTEM\CurrentControlSet\Services\Eventlog\'+$LogName) CustomSD $newSDDL;

Source-Related

Sources are actually manipulated using the registry (not through some API), and therefore require actual permissions on the registry key of the log itself. For instance, to create a new event source in the Application log, I would need the privilege to create a key under HKLM\SYSTEM\CurrentControlSet\services\eventlog\Application.
Note: Consider creating all of the event sources in one concentrated blow as an admin, to avoid messing with the registry's permissions.
No restart is required to apply these permissions.

Scripting Time

Granting / denying privileges on the registry is fairly easy with PowerShell. Here's a bit that adds read/write permissions for sources on a specific log to a specific principle:

# Set parameters:
param(
[string]$Principle = 'Builtin\Remote Desktop Users',
[string]$LogName = 'Application'
)

# Compose Key:
$LogPath = 'HKLM:\SYSTEM\CurrentControlSet\services\eventlog\'+$LogName;
if(Test-Path $LogPath)
{
    $acl = Get-Acl $LogPath
    $ace = New-Object System.Security.AccessControl.RegistryAccessRule $Principle,'WriteKey, ReadKey','allow'
    $acl.AddAccessRule($ace)
    #Set-Acl $LogPath $acl
}else{Write-Error "Cannot acesss log $LogName"}

Further Reading

Enjoy messing with the event log!