Here is a sed solution:
$ echo ':29.06.2019 23:03:17' | sed 's/^://'
29.06.2019 23:03:17
What the command sed 's/^://' is doing is substitute s the colon character : from the beginning ^ of each line with the empty string //.
Here is a tricky awk solution, where we changing the field separator to ^:, described above, and output the second field (of each line):
$ echo ':29.06.2019 23:03:17' | awk -F'^:' '{print $2}'
29.06.2019 23:03:17
The task could be accomplished also with grep (explanation), probably this could be the fastest solution for large amount of data:
$ echo 'Logfile started :29.06.2019 23:03:17' | grep -Po '^Logfile started :\K.*'
29.06.2019 23:03:17
Or process the file directly by the following command, where the limitation ^ is removed:
grep -Po 'Logfile started :\K.*' process.log
The above could be achieved also by sed and capture groups ()->\1:
sed -nr 's/^.*Logfile started :(.*)$/\1/p' process.log
Where the expression ^.*<something>.*$ will match the whole line, that contains <something>. The command s/old/new/ will substitute this line by the content of the first capture group (the expression in the brackets could be more concrete). The option -r enables the extended regular expressions. The option -n will suppress the normal output of sed and finally the command p will print the matches.